Difference between revisions of "MAT2233"

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==Topics List==
 
==Topics List==
 
{| class="wikitable sortable"
 
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
+
! Date !! Sections from Lay !! Sections from Bretscher !! Topics !! Prerequisite Skills !! Student Learning Outcomes
  
 
|-   
 
|-   
 +
  
 
|Week 1
 
|Week 1
Line 14: Line 15:
 
||
 
||
  
<div style="text-align: center;">1.1, 1.2</div>
+
<div style="text-align: center;">1.1 and 1.2</div>
 +
 
 +
||
 +
 
 +
<div style="text-align: center;">1.1</div>
  
 
||
 
||
 
          
 
          
[[Systems of Linear Equations]]  
+
[[Introduction to Linear Systems of Equations]]  
  
 
||
 
||
  
* Adding and subtracting equations
+
* Adding equations and multiplying equations by constants <!-- 1073-Mod 12.1 --> 
* Solving an equation for a specified variable
+
* [[Solving Equations and Inequalities]] <!-- 1073-Mod R --> 
  
 
||
 
||
  
* Vectors and Matrices
+
* Using elimination to find solutions of linear systems
* Gauss-Jordan elimination
+
* The Geometrical interpretation of solutions to linear systems
  
 
|-
 
|-
Line 37: Line 42:
 
||
 
||
  
<div style="text-align: center;">1.3</div>
+
<div style="text-align: center;">1.3, 1.4, and 1.5</div>
 +
 
 +
||
 +
 
 +
<div style="text-align: center;">1.2 and 1.3</div>
  
 
||
 
||
 
          
 
          
[[Solutions of Linear Systems]]  
+
[[Vectors and Matrices]]
 +
 
 +
[[Gauss-Jordan Elimination]]  
  
 
||
 
||
  
* Gauss-Jordan elimination
+
* [[Introduction to Linear Systems of Equations]] <!-- 2233-1.1 & 1.2 --> 
* Equation for a line
+
* [[Linear Equations|Equation for a line]] <!-- 1073-Mod R --> 
  
 
||
 
||
  
* Rank of a matrix
+
* Vectors and vector addition
* Matrix addition
+
* Matrix notation
* The product Ax
+
* The Gauss-Jordan method for solving a linear system of equation
* Inner product
+
* 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
 
* Linear Combinations
  
  
 
|-
 
|-
 +
 +
 +
|Week&nbsp;3
 +
 +
||
 +
 +
<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]] <!-- 1073-Mod 1.2-> 
 +
* [[Vectors and Matrices]], [[Gauss-Jordan Elimination]]  <!-- 2233-1.3--> 
 +
* [[Transformations of Functions]]  <!-- 1073-Mod 6 --> 
 +
 +
||
 +
 +
* Matrix Operations
 +
* Matrix products by columns
 +
* Matrix products using the dot product
 +
 +
|-
 +
  
  
Line 64: Line 108:
 
||
 
||
  
<div style="text-align: center;">2.1, 2.2, 2.3, and 2.4</div>
+
<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]]  
[[Linear Transformations]]  
 
  
 
||
 
||
  
* Domain and Range
+
* [[Matrix Algebra and Matrix Multiplication]]  <!-- 2233-2.3-->
* Matrix addition
+
* [[Inverse functions and the identity function|Inverse Functions]] <!-- 1073-7.2-->
* Rotations and scaling in geometry
+
* [[Introduction to Linear Systems of Equations]] <!-- 2233-1.1 & 1.2 --> 
  
 
||
 
||
  
* Linear transformations and their properties
+
* The Identity matrix
* Geometry of Linear Transformations (rotations, scalings and projections)
+
* The Inverse of a Matrix
 +
* Various characterizations for an invertible matrix
 +
 
  
 
|-
 
|-
 +
 +
  
  
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||
 
||
  
<div style="text-align: center;">2.1, 2.2, 2.3, and 2.4</div>
+
<div style="text-align: center;">1.8 and 1.9</div>
 +
 
 +
||
 +
 
 +
<div style="text-align: center;">2.1</div>
  
 
||
 
||
 
    
 
    
 
+
[[Introduction to Linear Transformations]]  
[[Matrix Products]]  
 
  
 
||
 
||
  
* Linear Combinations
+
* [[Introduction to Linear Systems of Equations]] <!-- 2233-1.1 & 1.2 -->
* Inverse functions and the identity function
+
* [[Vectors and Matrices]], [[Gauss-Jordan Elimination]]  <!-- 1073-7.2-->
* Vectors and the Inner product
+
* [[Transformations of Functions]]  <!-- 1073-Mod 6 --> 
  
 
||
 
||
  
* Matrix Products (both inner product and row-by-column methods)
+
* Linear Transformations
* The Inverses of a linear transform
+
* Requirements for a transformation to be linear
 
 
  
 
|-
 
|-
  
 +
|Week&nbsp;5 
  
 +
||
  
|Week&nbsp;5
+
<div style="text-align: center;">1.7, 2.8, and 2.9</div>
  
 
||
 
||
  
<div style="text-align: center;">3.1, 3.2, and 3.3</div>
+
<div style="text-align: center;">3.2</div>
  
 
||
 
||
 
    
 
    
[[Subspaces in Different Dimensions]]  
+
[[Subspaces of Rⁿ and Linear Independence]]  
  
 
||
 
||
  
* Image and kernel of a function
+
* [[Matrix Algebra and Matrix Multiplication]]  <!-- 2233-2.3-->
* Linear transformations
+
* [[The Inverse of a Linear Transformation]]
  
 
||
 
||
  
* Image and Kernel of a linear transformation
+
* Definition of a subspace of Rⁿ
* Span of a vector set
+
* Defining linear independence for a set of vectors
* Subspace of R<sup>n</sup>
+
* Definition of a basis for a subspace
  
  
 
|-
 
|-
  
 +
|Week&nbsp;6 
 +
 +
||
  
|Week&nbsp;6
+
<div style="text-align: center;">4.1</div>
  
 
||
 
||
  
<div style="text-align: center;">3.1, 3.2, and 3.3</div>
+
<div style="text-align: center;">4.1</div>
  
 
||
 
||
 
    
 
    
[[Bases and Linear Independence]]  
+
[[Introduction to Vector Spaces]]
  
 
||
 
||
  
* Linear Combinations
+
* [[Matrix Algebra and Matrix Multiplication]]  <!-- 2233-2.3-->  
* Dimension in R<sup>n</sup>
+
* [[Subspaces of Rⁿ and Linear Independence]]
* Rank of a matrix
 
* Subspace of R<sup>n</sup>
 
  
 
||
 
||
  
* Linear independence and basis
+
* Definition of a vector space(or linear space)
* Dimension
+
* Subspaces of vector spaces
* Rank-nullity Theorem
+
* Linear combinations and bases for vector spaces
 +
* Examples of vector spaces of functions
  
  
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|Week&nbsp;7/8  
+
|Week&nbsp;6  
  
 
||
 
||
  
<div style="text-align: center;"> 3.</div>
+
<div style="text-align: center;">4.2</div>
  
 
||
 
||
  
 +
<div style="text-align: center;">3.1</div>
 +
 +
||
 
    
 
    
[[Similar Matrices and Coordinates]]  
+
[[The Column Space and Nullspace of a Linear Transformation]]  
  
 
||
 
||
  
* Conics (ellipses in particular)
+
* [[Introduction to Linear Transformations ]]
* Equivalence Relations
+
* [[Range of a Function]] <!-- 1073- mod 1.2-->
 +
* [[The Inverse of a Linear Transformation]]
  
 
||
 
||
  
* Coordinates in a subspace of Rn
+
* The image (or column space) of a linear transformation
* Similar matrices
+
* The kernel (or nullspace) of a linear transformation
* Diagonal matrices
+
* Properties of the kernel
 +
 
 +
 
 +
|-
 +
 
 +
 
 +
|Week&nbsp;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]]
  
|Week&nbsp;9
 
  
 
||
 
||
  
<div style="text-align: center;"> 5.1, 5.2, 5.3, and 5.4 </div>
+
* 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&nbsp;8
  
 
||
 
||
  
 
+
<div style="text-align: center;">6.1 and 6.2</div>
[[Orthogonality]]
 
  
 
||
 
||
  
* Parallel and perpendicular lines
+
<div style="text-align: center;">Appendix A and 5.1</div>
* Absolute value function
 
* Basic trigonometric function
 
* Properties of inner products
 
  
 
||
 
||
 +
       
 +
[[Dot Products and Orthogonality]]
  
* Transpose of a Matrix
+
||
* Cauchy-Schwarz Inequality
 
* Orthonormal vectors
 
* Orthogonal complement
 
* Orthogonal Projection
 
* Orthonormal Bases
 
  
 +
* [[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&nbsp;10
+
|- 
 +
 +
 
 +
|Week&nbsp;9
  
 
||
 
||
  
<div style="text-align: center;"> 5.1, 5.2, 5.3, and 5.4 </div>
+
<div style="text-align: center;">6.3 and 6.4</div>
  
 
||
 
||
  
 
+
<div style="text-align: center;">5.2 and 5.3</div>
[[Gram-Schmidt Process]]  
+
 
 +
||
 +
       
 +
<p> [[Orthonormal Bases and the Gram-Schmidt Process]] </p>
 +
<p> [[Orthogonal Transformations and Orthogonal Matrices]] </p>
  
 
||
 
||
  
* Unit vectors
+
* [[Subspaces of Rⁿ and Linear Independence]] 
* Inner products
+
* [[Dot Products and Orthogonality]]
* Orthonormal bases
 
* Subspaces of R<big>n</big>
 
  
 
||
 
||
  
* Gram-Schmidt process
+
* Orthogonal transformations
* The Least Squares solution
+
* Orthonormal Bases
 +
* Orthogonal matrices
 +
* The transpose of a matrix
 +
* The Gram-Schmidt Process
 +
* QR factorization
 +
 
 +
|- 
 +
 +
 
 +
|Week&nbsp;10
  
 
||
 
||
  
 +
<div style="text-align: center;">6.5 and 6.6</div>
  
|-
+
||
  
 +
<div style="text-align: center;">5.4</div>
  
|Week&nbsp;11
+
||
 +
       
 +
[[The Least-squares Solution]]
  
 
||
 
||
  
<div style="text-align: center;">6.1, 6.2, and 6.3 </div>
+
* [[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 kernel of the transpose) for all matrices
[[Determinants]]
+
* The least-squares solution for a linear system
 +
* Data fitting using the least-squares solution
 +
 
 +
 
 +
|-
 +
 
 +
 
 +
 
 +
|Week&nbsp;11
 +
 
 +
||
 +
 
 +
<div style="text-align: center;">3.1 and 3.2</div>
  
 
||
 
||
  
* Summation notation
+
<div style="text-align: center;">6.1 and 6.2</div>
* Sgn function
 
  
 
||
 
||
 +
       
 +
<p> [[Introduction to Determinants]] </p>
 +
<p> [[Cramer's Rule]] </p>
  
* Properties of Determinants
+
||
* Sarrus's Rule
 
* Row operations and determinants
 
* Invertibility based on determinant
 
  
 +
* [[Orthonormal Bases and the Gram-Schmidt Process]] 
 +
* [[The Inverse of a Linear Transformation]]
 +
* [[Sigma 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
 +
  
  
Line 288: Line 420:
  
  
|Week&nbsp;12  
+
|Week&nbsp;12
  
 
||
 
||
  
<div style="text-align: center;">6.1, 6.2, and 6.3 </div>
+
<div style="text-align: center;">3.3</div>
  
 
||
 
||
  
 
+
<div style="text-align: center;">6.3</div>
[[Cramer's Rule]]
 
  
 
||
 
||
 +
       
 +
[[The Geometric Interpretation of the Determinant]]
  
* Properties of Determinants
+
||
* linear Systems
+
* Invertible matrices
+
* [[Orthonormal Bases and the Gram-Schmidt Process]] 
 +
* [[Introduction to Determinants]]
 +
* [[The Inverse of a Linear Transformation]]
  
 
||
 
||
  
* Parrallelepipeds in R<big>n</big>
 
* Geometric Interpretation of the Determinant
 
* Cramer's rule
 
  
||
+
* Cramer's Rule
 +
* The adjoint and inverse of a matrix
 +
* The area of a parallelogram and the volume of a parallelepiped
  
  
Line 317: Line 451:
  
  
|Week&nbsp;13  
+
|Week&nbsp;13
  
 
||
 
||
  
<div style="text-align: center;">7.1, 7.2, 7.3, and 8.1 </div>
+
<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]]  
 
[[Eigenvalues and Eigenvectors]]  
  
 
||
 
||
  
* Finding real roots of a polynomial
+
* [[Introduction to Determinants]]   
* Finding the kernel of a function
+
* [[The Column Space and Nullspace of a Linear Transformation]]
 +
* [[The Inverse of a Linear Transformation]]
  
 
||
 
||
  
* Diagonalization
+
* The requirement for a matrix to be diagonalizable
* Finding eigenvalues
+
* Definition of an eigenvector
* Finding eigenvectors
+
* The characteristic equation used to find eigenvalues
* Geometric and algebraic multiplicity
+
* Eigenvalues of a triangular matrix
 +
* Eigenspaces for specific eigenvalues
  
||
 
  
 +
|-
  
|-
 
  
|Week&nbsp;14  
+
|Week&nbsp;14
  
 
||
 
||
  
<div style="text-align: center;">7.1, 7.2, 7.3, and 8.1 </div>
+
<div style="text-align: center;">5.3 and 5.4</div>
  
 
||
 
||
  
 
+
<div style="text-align: center;">3.4 and 7.3</div>
[[Spectral Theorem]]  
+
 
 +
||
 +
       
 +
[[Diagonalization of Matrices]]  
  
 
||
 
||
  
* Transpose of a matrix
+
* [[Eigenvalues and Eigenvectors]] 
* Basis
+
* [[The Column Space and Nullspace of a Linear Transformation]]
* Orthogonal matrices
 
* Diagonal matrices
 
  
 
||
 
||
  
* Symmetric matrices
+
* Similar matrices
* Spectral Theorem
+
* Diagonalization in terms of linearly independent eigenvectors
 +
* Algebraic and geometric multiplicity for a specific eigenvalue
 +
* The strategy for diagonalization
 +
 
  
||
+
 
 +
|-

Latest revision as of 12:58, 29 January 2022

A comprehensive list of all undergraduate math courses at UTSA can be found here.

The Wikipedia summary of Linear Algebra and its history.

Topics List

Date Sections from Lay Sections from Bretscher Topics Prerequisite Skills Student Learning Outcomes
Week 1
1.1 and 1.2
1.1

Introduction to Linear Systems of Equations

  • Using elimination to find solutions of linear systems
  • The Geometrical interpretation of solutions to linear systems
Week 2
1.3, 1.4, and 1.5
1.2 and 1.3

Vectors and Matrices

Gauss-Jordan Elimination

  • Vectors and vector addition
  • 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
2.1
2.3

Matrix Algebra and Matrix Multiplication

  • Matrix Operations
  • Matrix products by columns
  • Matrix products using the dot product
Week 3
2.2 and 2.3
2.4

The Inverse of a Linear Transformation

  • The Identity matrix
  • The Inverse of a Matrix
  • Various characterizations for an invertible matrix


Week 4
1.8 and 1.9
2.1

Introduction to Linear Transformations

  • Linear Transformations
  • Requirements for a transformation to be linear
Week 5
1.7, 2.8, and 2.9
3.2

Subspaces of Rⁿ and Linear Independence

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


Week 6
4.1
4.1

Introduction to Vector Spaces

  • 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
4.2
3.1

The Column Space and Nullspace 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
4.3 and 4.5
3.3 and 4.1

The Dimension of a Vector Space


  • The number of vectors in a basis of Rn
  • 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
6.1 and 6.2
Appendix A and 5.1

Dot Products and Orthogonality

  • 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
6.3 and 6.4
5.2 and 5.3

Orthonormal Bases and the Gram-Schmidt Process

Orthogonal Transformations and Orthogonal Matrices

  • Orthogonal transformations
  • Orthonormal Bases
  • Orthogonal matrices
  • The transpose of a matrix
  • The Gram-Schmidt Process
  • QR factorization
Week 10
6.5 and 6.6
5.4

The Least-squares Solution

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


Week 11
3.1 and 3.2
6.1 and 6.2

Introduction to Determinants

Cramer's Rule

  • 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
3.3
6.3

The Geometric Interpretation of the Determinant


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


Week 13
The beginning of 5.3 as well as the sections 5.1 and 5.2
7.1, 7.2 and the beginning of 7.3

Eigenvalues and Eigenvectors

  • 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
5.3 and 5.4
3.4 and 7.3

Diagonalization of Matrices

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