Department of Mathematics at UTSA - User contributions [en]

MAT2233

2020-09-15T19:48:34Z

Nikos.salingaros:

MAT2233

2020-09-15T19:43:26Z

Nikos.salingaros:

MAT2233

2020-07-30T17:04:14Z

Nikos.salingaros:

A comprehensive list of all undergraduate math courses at UTSA can be found [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ here].

The Wikipedia summary of [https://en.wikipedia.org/wiki/Linear_algebra Linear Algebra and its history].

==Topics List==
{| class="wikitable sortable"
! Date !! Sections from Lay !! Sections from Bretscher !! Topics !! Prerequisite Skills !! Student Learning Outcomes

|-

|Week 1

||

<div style="text-align: center;">1.1 and 1.2</div>

||

<div style="text-align: center;">1.1</div>

||

[[Introduction to Linear Systems of Equations]]

||

* [[Systems of Equations in Two Variables| Adding equations and multiplying equations by constants]] 
* [[Solving Equations]] 

||

* Using elimination to find solutions of linear systems
* The Geometrical interpretation of solutions to linear systems

|-

|Week 2

||

<div style="text-align: center;">1.3, 1.4, and 1.5</div>

||

<div style="text-align: center;">1.2 and 1.3</div>

||

[[Vectors, Matrices, and Gauss-Jordan Elimination]]

||

* [[Introduction to Linear Systems of Equations]] 
* [[Linear Equations|Equation for a line]] 

||

* Vectors and vector spaces
* Matrix notation
* The Gauss-Jordan method for solving a linear system of equation
* The rank of a matrix
* Sums of Matrices
* The product Ax (where A is a matrix and x is a vector)
* The Dot product
* Linear Combinations

|-

|Week 3

||

<div style="text-align: center;">1.8 and 1.9</div>

||

<div style="text-align: center;">2.1</div>

||

[[Introduction to Linear Transformations]]

||

* [[Introduction to Linear Systems of Equations]] 
* [[Vectors, Matrices, and Gauss-Jordan Elimination]] 
* [[Transformations of Functions]] 

||

* Linear Transformation
* Requirements for a transformation to be linear

|-

|Week 4

||

<div style="text-align: center;">2.1</div>

||

<div style="text-align: center;">2.3</div>

||

[[Matrix Algebra and Matrix Multiplication]]

||

* [[Range of a Function]] 
* [[Transformations of Functions]] 

||

* Matrix Operations
* Matrix products by columns
* Matrix products using the dot product

|-

|Week 4

||

<div style="text-align: center;">2.2 and 2.3</div>

||

<div style="text-align: center;">2.4</div>

||

[[The Inverse of a Linear Transformation]]

||

* [[Matrix Algebra and Matrix Multiplication]] 
* [[Inverse functions and the identity function|Inverse Functions]] 
* [[Introduction to Linear Systems of Equations]] 

||

* The Identity matrix
* The Inverse of a Matrix
* The Inverse of a Linear Transformation
* Various characterizations for an invertible matrix

|-

|Week 5

||

<div style="text-align: center;">1.7, 2.8, and 2.9</div>

||

<div style="text-align: center;">3.2</div>

||

[[Subspaces of Rⁿ and Linear Independence]]

||

* [[Matrix Algebra and Matrix Multiplication]] 
* [[The Inverse of a Linear Transformation]]

||

* Definition of a subspace of Rⁿ
* Defining linear independence for a set of vectors
* Definition of a basis for a subspace

|-

|Week 6

||

<div style="text-align: center;">4.1</div>

||

<div style="text-align: center;">4.1</div>

||

[[Introduction to Vector Spaces]]

||

* [[Matrix Algebra and Matrix Multiplication]] 
* [[Subspaces of Rⁿ and Linear Independence]]

||

* Definition of a vector space(or linear space)
* Subspaces of vector spaces
* Linear combinations and bases for vector spaces
* Examples of vector spaces of functions

|-

|Week 6

||

<div style="text-align: center;">4.2</div>

||

<div style="text-align: center;">3.1</div>

||

[[The Column Space and Nullspace of a Linear Transformation]]

||

* [[Introduction to Linear Transformations ]]
* [[Range of a Function]] 
* [[The Inverse of a Linear Transformation]]

||

* The image (or column space) of a linear transformation
* The kernel (or nullspace) of a linear transformation
* Properties of the kernel

|-

|Week 7

||

<div style="text-align: center;">4.3 and 4.5</div>

||

<div style="text-align: center;">3.3 and 4.1</div>

||

[[The Dimension of a Vector Space]]

||

* [[Introduction to Vector Spaces]]
* [[Subspaces of Rⁿ and Linear Independence]]

||

* The number of vectors in a basis of R<sup>n</sup>
* Dimension of a subspace in Rⁿ
* The dimension of a vector space
* The dimension of the nullspace (or kernel) and the column space (or image)
* The Rank-nullity Theorem

|-

|-

|Week 8

||

<div style="text-align: center;">6.1 and 6.4</div>

||

<div style="text-align: center;">Appendix A and 5.1</div>

||

[[Dot Products and Orthogonality]]

||

* [[The Dimension of a Vector Space]]
* [[Subspaces of Rⁿ and Linear Independence]]

||

* Orthogonal vectors
* Length (or magnitude or norm) of a vector
* Unit vectors
* Orthonormal vectors
* Orthogonal projections
* Orthogonal complements
* Cauchy-Schwarz inequality
* The angle between vectors

|-

|-

|Week 9

||

<div style="text-align: center;">6.3 and 6.4</div>

||

<div style="text-align: center;">5.2 and 5.3</div>

||

[[Orthonormal Bases and the Gram-Schmidt Process]]

||

* [[Subspaces of Rⁿ and Linear Independence]]
* [[Dot Products and Orthogonality]]

||

* Orthogonal transformations
* Orthonormal Bases
* Orthogonal matrices
* The transpose of a matrix
* The Gram-Schmidt Process
* QR factorization

|-

|Week 10

||

<div style="text-align: center;">6.5 and 6.6</div>

||

<div style="text-align: center;">5.4</div>

||

[[The Least-squares Solution]]

||

* [[Dot Products and Orthogonality]]
* [[The Column Space and Nullspace of a Linear Transformation]]

||

* The orthogonal complement of the image is equal to the left nullspace (or kernal of the transpose) for all matrices
* The least-squares solution for a linear system
* Data fitting using the least-squares solution

|-

|Week 11

||

<div style="text-align: center;">3.1 and 3.2</div>

||

<div style="text-align: center;">6.1 and 6.2</div>

||

[[Introduction to Determinants]]

||

* [[Orthonormal Bases and the Gram-Schmidt Process]]
* [[The Inverse of a Linear Transformation]]
* '''[[Signum Notation]]'''

||

* The determinant of 2 by 2 and 3 by 3 matrices
* The determinant of a general n by n matrix
* The determinant of a triangular matrix
* Properties of the determinant
* The determinant of the transpose
* Invertibility and the determinant

|-

|Week 12

||

<div style="text-align: center;">3.3</div>

||

<div style="text-align: center;">6.3</div>

||

[[The Geometric Interpretation of the Determinant]]

||

* [[Orthonormal Bases and the Gram-Schmidt Process]]
* [[Introduction to Determinants]]
* [[The Inverse of a Linear Transformation]]

||

* Cramer's Rule
* The adjoint and inverse of a matrix
* The area of a parallelogram and the volume of a parallelepiped

|-

|Week 13

||

<div style="text-align: center;">The beginning of 5.3 as well as the sections 5.1 and 5.2</div>

||

<div style="text-align: center;">7.1, 7.2 and the beginning of 7.3</div>

||

[[Eigenvalues and Eigenvectors]]

||

* [[Introduction to Determinants]]
* [[The Column Space and Nullspace of a Linear Transformation]]
* [[The Inverse of a Linear Transformation]]

||

* The requirement for a matrix to be diagonalizable
* Definition of an eigenvector
* The characteristic equation used to find eigenvalues
* Eigenvalues of a triangular matrix
* Eigenspaces for specific eigenvalues

|-

|Week 14

||

<div style="text-align: center;">5.3 and 5.4</div>

||

<div style="text-align: center;">3.4 and 7.3</div>

||

[[Diagonalization of Matrices]]

||

* [[Eigenvalues and Eigenvectors]]
* [[The Column Space and Nullspace of a Linear Transformation]]

||

* Similar matrices
* Diagonalization in terms of linearly independent eigenvectors
* Algebraic and geometric multiplicity for a specific eigenvalue
* The strategy for diagonalization

|-

|-

{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes

|-

|Week 1

||

<div style="text-align: center;">1.1, 1.2</div>

||

[[Systems of Linear Equations]]

||

* [[Systems of Equations in Two Variables| Adding and multiplying equations by constants]] 
* [[Solving Equations]] 

||

* Vectors and Matrices
* Gauss-Jordan elimination

|-

|Week 2

||

<div style="text-align: center;">1.3</div>

||

[[Solutions of Linear Systems]]

||

* [[Systems of Linear Equations|Gauss-Jordan elimination]] 
* [[Linear Equations|Equation for a line]] 

||

* Rank of a matrix
* Matrix addition
* The product Ax (where A is a matrix and x is a vector)
* The Inner product
* Linear Combinations

|-

|Week 3

||

<div style="text-align: center;">2.1 and 2.2</div>

||

[[Linear Transformations]]

||

* [[Range of a Function]] 
* [[Transformations of Functions]] 

||

* Linear transformations and their properties
* Geometry of Linear Transformations (rotations, scalings and projections)

|-

|Week 4

||

<div style="text-align: center;"> 2.3 and 2.4</div>

||

[[Matrix Products and Inverses]]

||

* [[Solutions of Linear Systems| Linear Combinations]] 
* [[Inverse functions and the identity function|Inverse Functions]] 
* [[Solutions of Linear Systems|Vectors and the Inner product]] 

||

* Matrix Products (both inner product and row-by-column methods)
* The Inverses of a linear transform

|-

|Week 6

||

<div style="text-align: center;">3.1</div>

||

[[Image and Kernel of a Linear Transform]]

||

* [[Solutions of Linear Systems]] 
* [[Range of a Function|Image of a Function]] 
* [[Kernel of a Function]] 

||

* The image of a Linear transformation
* The kernel of a linear transformation
* Span of a set of vectors
* Alternative characterizations of Invertible matrices

|-

|Week 6

||

<div style="text-align: center;">3.2</div>

||

[[Linear Independence]]

||

* [[Solutions of Linear Systems]] 
* [[Image and Kernel of a Linear Transform]] 

||

* Subspaces of Rⁿ
* Redundant vectors and linear independence
* Characterizations of Linear Independence

|-

|Week 6

||

<div style="text-align: center;">3.2</div>

||

[[Bases of Subspaces]]

||

* [[Linear Independence]] 
* [[Image and Kernel of a Linear Transform|The span of a set of vectors]] 

||

* Bases and Linear independence
* Basis of the image
* Basis and unique representation

|-

|Week 5

||

<div style="text-align: center;">3.3</div>

||

[[The Dimension of a Subspace]]

||

* [[Range of a Function|Image of a Function]] 
* [[Bases and Linear Independence]] 
* [[Linear transformations]] 

||

* Dimension of the Image
* Rank-nullity theorem
* Various bases in R<sup>n</sup>

|-

|Week 7/8

||

<div style="text-align: center;"> 3.4 </div>

||

[[Similar Matrices and Coordinates]]

||

* [[Bases of Subspaces]] 
* '''[[Equivalence Relations]]''' 

||

* Coordinates in a subspace of Rⁿ
* Similar matrices
* Diagonal matrices

||

|-

|Week 9

||

<div style="text-align: center;"> 5.1</div>

||

[[Orthogonal Projections and Orthonormal Bases]]

||

* [[Parallel and Perpendicular Lines]] 
* [[Absolute value function]]
* [[Trig. Functions: Unit Circle Approach]] 
* [[Matrix Products and Inverses|Inner Products]] 
* [[Bases and Linear Independence]] 

||

* Magnitude (or norm or length) of a vector
* Unit Vectors
* Cauchy-Schwarz Inequality
* Orthonormal vectors
* Orthogonal complement
* Orthogonal Projection
* Orthonormal bases
* Angle between vectors

||

|-

|Week 10

||

<div style="text-align: center;">5.2 </div>

||

[[Gram-Schmidt Process and QR Factorization]]

||

* [[Orthogonal Projections and Orthonormal Bases|Unit vectors]] 
* [[Matrix Products and Inverses|Inner Products]] 
* [[Orthogonal Projections and Orthonormal Bases|Orthonormal Bases]] 
* [[Bases and Linear Independence]] 

||

* Gram-Schmidt process
* QR Factorization

||

|-

|Week 11

||

<div style="text-align: center;">5.3</div>

||

[[Orthogonal Transformations and Orthogonal Matrices]]

||

* [[Image and Kernel of a Linear Transform]] 
* [[Matrix Products and Inverses|Inner Products]] 
* [[Orthogonal Projections and Orthonormal Bases]]

||

* Orthogonal Transformations
* Properties of Othogonal Transformations
* Transpose of a Matrix
* The matrix of an Orthogonal Projection

||

|-

|Week 11

||

<div style="text-align: center;">5.3</div>

||

[[Least Squares]]

||

* [[Linear transformations]] 
* [[Orthogonal Transformations and Orthogonal Matrices]] 
* [[Orthogonal Projections and Orthonormal Bases|Orthogonal Projections]] 

||

* The Least Squares Solution
* The Normal Equation
* Another matrix for an Orthogonal Projection

||

|-
|Week 11

||

<div style="text-align: center;">6.1 and 6.2</div>

||

[[Determinants]]

||

* [[Summation Notation]] 
* [[Sgn Function]] 
* [[Matrix Products and Inverses|Inverse of a Linear Transformation]] 
* [[Orthogonal Transformations and Orthogonal Matrices| Transpose of a Matrix]] 

||

* Properties of Determinants
* Sarrus's Rule
* Row operations and determinants
* Invertibility based on the determinant

||

|-

|Week 12

||

<div style="text-align: center;">6.3 </div>

||

[[Cramer's Rule]]

||

* [[Determinants]] 
* [[Matrix Products and Inverses| Invertible matrices]] 
* [[Linear Transformations| Rotations]] 

||

* Parrallelepipeds in R<big>n</big>
* Geometric Interpretation of the Determinant
* Cramer's rule

||

|-

|Week 13

||

<div style="text-align: center;">7.1</div>

||

[[Diagonalization]]

||

* [[Similar Matrices and Coordinates]] 
* [[Orthogonal Transformations and Orthogonal Matrices]] 

||

* Diagonalizable matrices
* Eigenvalues and eigenvectors
* Real eigenvalues of orthogonal matrices

||

|-

|Week 14

||

<div style="text-align: center;">7.2 and 7.3</div>

||

[[Finding Eigenvalues and Eigenvectors]]

||

* [[Determinants]] 
* [[Matrix Products and Inverses| Invertible matrices]] 
* [[Diagonalization]] 
* [[Image and Kernel of a Linear Transform]] 

||

* Eigenvalues from the characteristic equation
* Eigenvalues of Triangular matrices
* Characteristic Polynomial
* Eigenspaces and eigenvectors
* Geometric and algebraic multiplicity
* Eigenvalues of similar matrices

|-

|Week 14

||

<div style="text-align: center;">8.1</div>

||

[[Symmetric Matrices]]

||

* [[Similar Matrices and Coordinates]] 
* [[Orthogonal Transformations and Orthogonal Matrices| Transpose of a Matrix]] 
* [[Diagonalization|Eigenvalues and Eigenvectors]] 
* [[Finding Eigenvalues and Eigenvectors|Algebraic and Geometric Multiplicities]] 

||

* Orthogonally Diagonalizable Matrices
* Spectral Theorem
* The real eigenvalues of a symmetric matrix

|-

|Week 14

||

<div style="text-align: center;">8.2</div>

||

[[Quadratic Forms]]

||

* [[Symmetric Matrices]] 
* [[Finding Eigenvalues and Eigenvectors]] 
* '''[[Conics]]''' 

||

* Quadratic Forms
* Diagonalizing a Quadratic Form
* Definiteness of a Quadratic Form
* '''Principal Axes''' 
* '''Ellipses and Hyperbolas from Quadratic Forms''' 

||

MAT2233

2020-07-30T17:03:27Z

Nikos.salingaros:

A comprehensive list of all undergraduate math courses at UTSA can be found [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ here].

The Wikipedia summary of [https://en.wikipedia.org/wiki/Linear_algebra Linear Algebra and its history].

==Topics List==
{| class="wikitable sortable"
! Date !! Sections from Lay !! Sections from Bretscher !! Topics !! Prerequisite Skills !! Student Learning Outcomes

|-

|Week 1

||

<div style="text-align: center;">1.1 and 1.2</div>

||

<div style="text-align: center;">1.1</div>

||

[[Introduction to Linear Systems of Equations]]

||

* [[Systems of Equations in Two Variables| Adding equations and multiplying equations by constants]] 
* [[Solving Equations]] 

||

* Using elimination to find solutions of linear systems
* The Geometrical interpretation of solutions to linear systems

|-

|Week 2

||

<div style="text-align: center;">1.3, 1.4, and 1.5</div>

||

<div style="text-align: center;">1.2 and 1.3</div>

||

[[Vectors, Matrices, and Gauss-Jordan Elimination]]

||

* [[Introduction to Linear Systems of Equations]] 
* [[Linear Equations|Equation for a line]] 

||

* Vectors and vector spaces
* Matrix notation
* The Gauss-Jordan method for solving a linear system of equation
* The rank of a matrix
* Sums of Matrices
* The product Ax (where A is a matrix and x is a vector)
* The Dot product
* Linear Combinations

|-

|Week 3

||

<div style="text-align: center;">1.8 and 1.9</div>

||

<div style="text-align: center;">2.1</div>

||

[[Introduction to Linear Transformations]]

||

* [[Introduction to Linear Systems of Equations]] 
* [[Vectors, Matrices, and Guass-Jordan Elimination]] 
* [[Transformations of Functions]] 

||

* Linear Transformation
* Requirements for a transformation to be linear

|-

|Week 4

||

<div style="text-align: center;">2.1</div>

||

<div style="text-align: center;">2.3</div>

||

[[Matrix Algebra and Matrix Multiplication]]

||

* [[Range of a Function]] 
* [[Transformations of Functions]] 

||

* Matrix Operations
* Matrix products by columns
* Matrix products using the dot product

|-

|Week 4

||

<div style="text-align: center;">2.2 and 2.3</div>

||

<div style="text-align: center;">2.4</div>

||

[[The Inverse of a Linear Transformation]]

||

* [[Matrix Algebra and Matrix Multiplication]] 
* [[Inverse functions and the identity function|Inverse Functions]] 
* [[Introduction to Linear Systems of Equations]] 

||

* The Identity matrix
* The Inverse of a Matrix
* The Inverse of a Linear Transformation
* Various characterizations for an invertible matrix

|-

|Week 5

||

<div style="text-align: center;">1.7, 2.8, and 2.9</div>

||

<div style="text-align: center;">3.2</div>

||

[[Subspaces of Rⁿ and Linear Independence]]

||

* [[Matrix Algebra and Matrix Multiplication]] 
* [[The Inverse of a Linear Transformation]]

||

* Definition of a subspace of Rⁿ
* Defining linear independence for a set of vectors
* Definition of a basis for a subspace

|-

|Week 6

||

<div style="text-align: center;">4.1</div>

||

<div style="text-align: center;">4.1</div>

||

[[Introduction to Vector Spaces]]

||

* [[Matrix Algebra and Matrix Multiplication]] 
* [[Subspaces of Rⁿ and Linear Independence]]

||

* Definition of a vector space(or linear space)
* Subspaces of vector spaces
* Linear combinations and bases for vector spaces
* Examples of vector spaces of functions

|-

|Week 6

||

<div style="text-align: center;">4.2</div>

||

<div style="text-align: center;">3.1</div>

||

[[The Column Space and Nullspace of a Linear Transformation]]

||

* [[Introduction to Linear Transformations ]]
* [[Range of a Function]] 
* [[The Inverse of a Linear Transformation]]

||

* The image (or column space) of a linear transformation
* The kernel (or nullspace) of a linear transformation
* Properties of the kernel

|-

|Week 7

||

<div style="text-align: center;">4.3 and 4.5</div>

||

<div style="text-align: center;">3.3 and 4.1</div>

||

[[The Dimension of a Vector Space]]

||

* [[Introduction to Vector Spaces]]
* [[Subspaces of Rⁿ and Linear Independence]]

||

* The number of vectors in a basis of R<sup>n</sup>
* Dimension of a subspace in Rⁿ
* The dimension of a vector space
* The dimension of the nullspace (or kernel) and the column space (or image)
* The Rank-nullity Theorem

|-

|-

|Week 8

||

<div style="text-align: center;">6.1 and 6.4</div>

||

<div style="text-align: center;">Appendix A and 5.1</div>

||

[[Dot Products and Orthogonality]]

||

* [[The Dimension of a Vector Space]]
* [[Subspaces of Rⁿ and Linear Independence]]

||

* Orthogonal vectors
* Length (or magnitude or norm) of a vector
* Unit vectors
* Orthonormal vectors
* Orthogonal projections
* Orthogonal complements
* Cauchy-Schwarz inequality
* The angle between vectors

|-

|-

|Week 9

||

<div style="text-align: center;">6.3 and 6.4</div>

||

<div style="text-align: center;">5.2 and 5.3</div>

||

[[Orthonormal Bases and the Gram-Schmidt Process]]

||

* [[Subspaces of Rⁿ and Linear Independence]]
* [[Dot Products and Orthogonality]]

||

* Orthogonal transformations
* Orthonormal Bases
* Orthogonal matrices
* The transpose of a matrix
* The Gram-Schmidt Process
* QR factorization

|-

|Week 10

||

<div style="text-align: center;">6.5 and 6.6</div>

||

<div style="text-align: center;">5.4</div>

||

[[The Least-squares Solution]]

||

* [[Dot Products and Orthogonality]]
* [[The Column Space and Nullspace of a Linear Transformation]]

||

* The orthogonal complement of the image is equal to the left nullspace (or kernal of the transpose) for all matrices
* The least-squares solution for a linear system
* Data fitting using the least-squares solution

|-

|Week 11

||

<div style="text-align: center;">3.1 and 3.2</div>

||

<div style="text-align: center;">6.1 and 6.2</div>

||

[[Introduction to Determinants]]

||

* [[Orthonormal Bases and the Gram-Schmidt Process]]
* [[The Inverse of a Linear Transformation]]
* '''[[Signum Notation]]'''

||

* The determinant of 2 by 2 and 3 by 3 matrices
* The determinant of a general n by n matrix
* The determinant of a triangular matrix
* Properties of the determinant
* The determinant of the transpose
* Invertibility and the determinant

|-

|Week 12

||

<div style="text-align: center;">3.3</div>

||

<div style="text-align: center;">6.3</div>

||

[[The Geometric Interpretation of the Determinant]]

||

* [[Orthonormal Bases and the Gram-Schmidt Process]]
* [[Introduction to Determinants]]
* [[The Inverse of a Linear Transformation]]

||

* Cramer's Rule
* The adjoint and inverse of a matrix
* The area of a parallelogram and the volume of a parallelepiped

|-

|Week 13

||

<div style="text-align: center;">The beginning of 5.3 as well as the sections 5.1 and 5.2</div>

||

<div style="text-align: center;">7.1, 7.2 and the beginning of 7.3</div>

||

[[Eigenvalues and Eigenvectors]]

||

* [[Introduction to Determinants]]
* [[The Column Space and Nullspace of a Linear Transformation]]
* [[The Inverse of a Linear Transformation]]

||

* The requirement for a matrix to be diagonalizable
* Definition of an eigenvector
* The characteristic equation used to find eigenvalues
* Eigenvalues of a triangular matrix
* Eigenspaces for specific eigenvalues

|-

|Week 14

||

<div style="text-align: center;">5.3 and 5.4</div>

||

<div style="text-align: center;">3.4 and 7.3</div>

||

[[Diagonalization of Matrices]]

||

* [[Eigenvalues and Eigenvectors]]
* [[The Column Space and Nullspace of a Linear Transformation]]

||

* Similar matrices
* Diagonalization in terms of linearly independent eigenvectors
* Algebraic and geometric multiplicity for a specific eigenvalue
* The strategy for diagonalization

|-

|-

{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes

|-

|Week 1

||

<div style="text-align: center;">1.1, 1.2</div>

||

[[Systems of Linear Equations]]

||

* [[Systems of Equations in Two Variables| Adding and multiplying equations by constants]] 
* [[Solving Equations]] 

||

* Vectors and Matrices
* Gauss-Jordan elimination

|-

|Week 2

||

<div style="text-align: center;">1.3</div>

||

[[Solutions of Linear Systems]]

||

* [[Systems of Linear Equations|Gauss-Jordan elimination]] 
* [[Linear Equations|Equation for a line]] 

||

* Rank of a matrix
* Matrix addition
* The product Ax (where A is a matrix and x is a vector)
* The Inner product
* Linear Combinations

|-

|Week 3

||

<div style="text-align: center;">2.1 and 2.2</div>

||

[[Linear Transformations]]

||

* [[Range of a Function]] 
* [[Transformations of Functions]] 

||

* Linear transformations and their properties
* Geometry of Linear Transformations (rotations, scalings and projections)

|-

|Week 4

||

<div style="text-align: center;"> 2.3 and 2.4</div>

||

[[Matrix Products and Inverses]]

||

* [[Solutions of Linear Systems| Linear Combinations]] 
* [[Inverse functions and the identity function|Inverse Functions]] 
* [[Solutions of Linear Systems|Vectors and the Inner product]] 

||

* Matrix Products (both inner product and row-by-column methods)
* The Inverses of a linear transform

|-

|Week 6

||

<div style="text-align: center;">3.1</div>

||

[[Image and Kernel of a Linear Transform]]

||

* [[Solutions of Linear Systems]] 
* [[Range of a Function|Image of a Function]] 
* [[Kernel of a Function]] 

||

* The image of a Linear transformation
* The kernel of a linear transformation
* Span of a set of vectors
* Alternative characterizations of Invertible matrices

|-

|Week 6

||

<div style="text-align: center;">3.2</div>

||

[[Linear Independence]]

||

* [[Solutions of Linear Systems]] 
* [[Image and Kernel of a Linear Transform]] 

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* Subspaces of Rⁿ
* Redundant vectors and linear independence
* Characterizations of Linear Independence

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|Week 6

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<div style="text-align: center;">3.2</div>

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[[Bases of Subspaces]]

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* [[Linear Independence]] 
* [[Image and Kernel of a Linear Transform|The span of a set of vectors]] 

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* Bases and Linear independence
* Basis of the image
* Basis and unique representation

|-

|Week 5

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<div style="text-align: center;">3.3</div>

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[[The Dimension of a Subspace]]

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* [[Range of a Function|Image of a Function]] 
* [[Bases and Linear Independence]] 
* [[Linear transformations]] 

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* Dimension of the Image
* Rank-nullity theorem
* Various bases in R<sup>n</sup>

|-

|Week 7/8

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<div style="text-align: center;"> 3.4 </div>

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[[Similar Matrices and Coordinates]]

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* [[Bases of Subspaces]] 
* '''[[Equivalence Relations]]''' 

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* Coordinates in a subspace of Rⁿ
* Similar matrices
* Diagonal matrices

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|-

|Week 9

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<div style="text-align: center;"> 5.1</div>

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[[Orthogonal Projections and Orthonormal Bases]]

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* [[Parallel and Perpendicular Lines]] 
* [[Absolute value function]]
* [[Trig. Functions: Unit Circle Approach]] 
* [[Matrix Products and Inverses|Inner Products]] 
* [[Bases and Linear Independence]] 

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* Magnitude (or norm or length) of a vector
* Unit Vectors
* Cauchy-Schwarz Inequality
* Orthonormal vectors
* Orthogonal complement
* Orthogonal Projection
* Orthonormal bases
* Angle between vectors

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|-

|Week 10

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<div style="text-align: center;">5.2 </div>

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[[Gram-Schmidt Process and QR Factorization]]

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* [[Orthogonal Projections and Orthonormal Bases|Unit vectors]] 
* [[Matrix Products and Inverses|Inner Products]] 
* [[Orthogonal Projections and Orthonormal Bases|Orthonormal Bases]] 
* [[Bases and Linear Independence]] 

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* Gram-Schmidt process
* QR Factorization

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|-

|Week 11

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<div style="text-align: center;">5.3</div>

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[[Orthogonal Transformations and Orthogonal Matrices]]

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* [[Image and Kernel of a Linear Transform]] 
* [[Matrix Products and Inverses|Inner Products]] 
* [[Orthogonal Projections and Orthonormal Bases]]

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* Orthogonal Transformations
* Properties of Othogonal Transformations
* Transpose of a Matrix
* The matrix of an Orthogonal Projection

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|-

|Week 11

||

<div style="text-align: center;">5.3</div>

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[[Least Squares]]

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* [[Linear transformations]] 
* [[Orthogonal Transformations and Orthogonal Matrices]] 
* [[Orthogonal Projections and Orthonormal Bases|Orthogonal Projections]] 

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* The Least Squares Solution
* The Normal Equation
* Another matrix for an Orthogonal Projection

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|-
|Week 11

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<div style="text-align: center;">6.1 and 6.2</div>

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[[Determinants]]

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* [[Summation Notation]] 
* [[Sgn Function]] 
* [[Matrix Products and Inverses|Inverse of a Linear Transformation]] 
* [[Orthogonal Transformations and Orthogonal Matrices| Transpose of a Matrix]] 

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* Properties of Determinants
* Sarrus's Rule
* Row operations and determinants
* Invertibility based on the determinant

||

|-

|Week 12

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<div style="text-align: center;">6.3 </div>

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[[Cramer's Rule]]

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* [[Determinants]] 
* [[Matrix Products and Inverses| Invertible matrices]] 
* [[Linear Transformations| Rotations]] 

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* Parrallelepipeds in R<big>n</big>
* Geometric Interpretation of the Determinant
* Cramer's rule

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|-

|Week 13

||

<div style="text-align: center;">7.1</div>

||

[[Diagonalization]]

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* [[Similar Matrices and Coordinates]] 
* [[Orthogonal Transformations and Orthogonal Matrices]] 

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* Diagonalizable matrices
* Eigenvalues and eigenvectors
* Real eigenvalues of orthogonal matrices

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|-

|Week 14

||

<div style="text-align: center;">7.2 and 7.3</div>

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[[Finding Eigenvalues and Eigenvectors]]

||

* [[Determinants]] 
* [[Matrix Products and Inverses| Invertible matrices]] 
* [[Diagonalization]] 
* [[Image and Kernel of a Linear Transform]] 

||

* Eigenvalues from the characteristic equation
* Eigenvalues of Triangular matrices
* Characteristic Polynomial
* Eigenspaces and eigenvectors
* Geometric and algebraic multiplicity
* Eigenvalues of similar matrices

|-

|Week 14

||

<div style="text-align: center;">8.1</div>

||

[[Symmetric Matrices]]

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* [[Similar Matrices and Coordinates]] 
* [[Orthogonal Transformations and Orthogonal Matrices| Transpose of a Matrix]] 
* [[Diagonalization|Eigenvalues and Eigenvectors]] 
* [[Finding Eigenvalues and Eigenvectors|Algebraic and Geometric Multiplicities]] 

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* Orthogonally Diagonalizable Matrices
* Spectral Theorem
* The real eigenvalues of a symmetric matrix

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|Week 14

||

<div style="text-align: center;">8.2</div>

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[[Quadratic Forms]]

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* [[Symmetric Matrices]] 
* [[Finding Eigenvalues and Eigenvectors]] 
* '''[[Conics]]''' 

||

* Quadratic Forms
* Diagonalizing a Quadratic Form
* Definiteness of a Quadratic Form
* '''Principal Axes''' 
* '''Ellipses and Hyperbolas from Quadratic Forms''' 

||