Difference between revisions of "MAT2233"

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||
 
||
  
<div style="text-align: center;">1.1, 1.2</div>
+
<div style="text-align: center;">1.1 and 1.2</div>
  
 
||
 
||
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||
 
||
  
* [[Systems of Equations in Two Variables| Adding and multiplying equations by constants]] <!-- 1073-Mod 12.1 -->   
+
* Adding equations and multiplying equations by constants <!-- 1073-Mod 12.1 -->   
* [[Solving Equations]] <!-- 1073-Mod R -->   
+
* [[Solving Equations and Inequalities]] <!-- 1073-Mod R -->   
  
 
||
 
||
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||
 
||
 
          
 
          
[[Vectors, Matrices, and Guass-Jordan Elimination]]  
+
[[Vectors and Matrices]]
 +
 
 +
[[Gauss-Jordan Elimination]]  
  
 
||
 
||
Line 59: Line 61:
 
||
 
||
  
* Vectors and vector spaces
+
* Vectors and vector addition
 
* Matrix notation
 
* Matrix notation
* The Guass-Jordan method for solving a linear system of equation
+
* The Gauss-Jordan method for solving a linear system of equation
 
* The rank of a matrix
 
* The rank of a matrix
 
* Sums of Matrices
 
* Sums of Matrices
Line 72: Line 74:
  
  
|Week&nbsp;3  
+
|Week&nbsp;3
  
 
||
 
||
  
<div style="text-align: center;">1.8 and 1.9</div>
+
<div style="text-align: center;">2.1</div>
  
 
||
 
||
  
<div style="text-align: center;">2.1</div>
+
<div style="text-align: center;">2.3</div>
  
 
||
 
||
 
    
 
    
[[Introduction to Linear Transformations]]  
+
[[Matrix Algebra and Matrix Multiplication]]  
  
 
||
 
||
  
* [[Introduction to Linear Systems of Equations]] <!-- 2233-1.1 & 1.2 -->  
+
* [[Range of a Function]] <!-- 1073-Mod 1.2->
* [[Vectors, Matrices, and Guass-Jordan Elimination]]  <!-- 1073-7.2-->  
+
* [[Vectors and Matrices]], [[Gauss-Jordan Elimination]]  <!-- 2233-1.3-->
 
* [[Transformations of Functions]]  <!-- 1073-Mod 6 -->   
 
* [[Transformations of Functions]]  <!-- 1073-Mod 6 -->   
  
 
||
 
||
  
* Linear Transformation
+
* Matrix Operations
* Requirements for a transformation to be linear
+
* Matrix products by columns
 +
* Matrix products using the dot product
 +
 
 +
|-
  
  
|-
 
  
|Week&nbsp;4    
+
|Week&nbsp;3    
  
 
||
 
||
  
<div style="text-align: center;">2.1</div>
+
<div style="text-align: center;">2.2 and 2.3</div>
  
 
||
 
||
  
<div style="text-align: center;">2.3</div>
+
<div style="text-align: center;">2.4</div>
  
 
||
 
||
 
    
 
    
[[Matrix Algebra and Matrix Multiplication]]  
+
[[The Inverse of a Linear Transformation]]  
  
 
||
 
||
  
* [[Range of a Function]] <!-- 1073-Mod 1.2->
+
* [[Matrix Algebra and Matrix Multiplication]] <!-- 2233-2.3-->  
* [[Vectors, Matrices, and Guass-Jordan Elimination]] <!-- 2233-1.3-->
+
* [[Inverse functions and the identity function|Inverse Functions]] <!-- 1073-7.2-->  
* [[Transformations of Functions]] <!-- 1073-Mod 6 -->   
+
* [[Introduction to Linear Systems of Equations]] <!-- 2233-1.1 & 1.2 -->   
  
 
||
 
||
  
* Matrix Operations
+
* The Identity matrix
* Matrix products by columns
+
* The Inverse of a Matrix
* Matrix products using the dot product
+
* Various characterizations for an invertible matrix
 +
 
  
 
|-
 
|-
 +
  
  
Line 134: Line 140:
 
||
 
||
  
<div style="text-align: center;">2.2 and 2.3</div>
+
<div style="text-align: center;">1.8 and 1.9</div>
  
 
||
 
||
  
<div style="text-align: center;">2.4</div>
+
<div style="text-align: center;">2.1</div>
  
 
||
 
||
 
    
 
    
[[The Inverse of a Linear Transformation]]  
+
[[Introduction to Linear Transformations]]  
  
 
||
 
||
  
* [[Matrix Algebra and Matrix Multiplication]] <!-- 2233-2.3-->  
+
* [[Introduction to Linear Systems of Equations]] <!-- 2233-1.1 & 1.2 -->  
* [[Inverse functions and the identity function|Inverse Functions]] <!-- 1073-7.2-->  
+
* [[Vectors and Matrices]], [[Gauss-Jordan Elimination]] <!-- 1073-7.2-->  
* [[Introduction to Linear Systems of Equations]] <!-- 2233-1.1 & 1.2 -->   
+
* [[Transformations of Functions]] <!-- 1073-Mod 6 -->   
  
 
||
 
||
  
* The Identity matrix
+
* Linear Transformations
* The Inverse of a Matrix
+
* Requirements for a transformation to be linear
* The Inverse of a Linear Transformation
 
* Various characterizations for an invertible matrix
 
 
 
  
 
|-
 
|-
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||
 
||
 
    
 
    
[[Subspaces of R^n and Linear Independence]]  
+
[[Subspaces of Rⁿ and Linear Independence]]  
  
 
||
 
||
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||
 
||
  
* Definition of a subspace of R<sup>n</sup>
+
* Definition of a subspace of Rⁿ
 
* Defining linear independence for a set of vectors
 
* Defining linear independence for a set of vectors
 
* Definition of a basis for a subspace
 
* Definition of a basis for a subspace
Line 205: Line 208:
  
 
* [[Matrix Algebra and Matrix Multiplication]]  <!-- 2233-2.3-->  
 
* [[Matrix Algebra and Matrix Multiplication]]  <!-- 2233-2.3-->  
* [[Subspaces of R^n and Linear Independence]]  
+
* [[Subspaces of Rⁿ and Linear Independence]]  
  
 
||
 
||
Line 266: Line 269:
  
 
* [[Introduction to Vector Spaces]]
 
* [[Introduction to Vector Spaces]]
* [[Subspaces of R<sup>n</sup> and Linear Independence]]  
+
* [[Subspaces of Rⁿ and Linear Independence]]  
  
  
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* The number of vectors in a basis of R<sup>n</sup>
 
* The number of vectors in a basis of R<sup>n</sup>
* Dimension of a subspace in R<sup>n</sup>
+
* Dimension of a subspace in Rⁿ
 
* The dimension of a vector space
 
* The dimension of a vector space
 
* The dimension of the nullspace (or kernel) and the column space (or image)
 
* The dimension of the nullspace (or kernel) and the column space (or image)
Line 280: Line 283:
 
|-
 
|-
  
 
 
|-
 
 
 
{| class="wikitable sortable"
 
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
 
  
 
|-   
 
|-   
 +
  
|Week&nbsp;1
+
|Week&nbsp;8
  
 
||
 
||
  
<div style="text-align: center;">1.1, 1.2</div>
+
<div style="text-align: center;">6.1 and 6.2</div>
 
 
||
 
       
 
[[Systems of Linear Equations]]
 
 
 
||
 
 
 
* [[Systems of Equations in Two Variables| Adding and multiplying equations by constants]] <!-- 1073-Mod 12.1 --> 
 
* [[Solving Equations]] <!-- 1073-Mod R --> 
 
 
 
||
 
 
 
* Vectors and Matrices
 
* Gauss-Jordan elimination
 
 
 
|-
 
 
 
 
 
|Week&nbsp;2
 
  
 
||
 
||
  
<div style="text-align: center;">1.3</div>
+
<div style="text-align: center;">Appendix A and 5.1</div>
  
 
||
 
||
 
          
 
          
[[Solutions of Linear Systems]]  
+
[[Dot Products and Orthogonality]]  
  
 
||
 
||
  
* [[Systems of Linear Equations|Gauss-Jordan elimination]] <!-- 2233-1.1 & 1.2 --> 
+
* [[The Dimension of a Vector Space]]  
* [[Linear Equations|Equation for a line]] <!-- 1073-Mod R --> 
+
* [[Subspaces of Rⁿ and Linear Independence]]  
  
 
||
 
||
  
* Rank of a matrix
+
* Orthogonal vectors
* Matrix addition
+
* Length (or magnitude or norm) of a vector
* The product Ax (where A is a matrix and x is a vector)
+
* Unit vectors
* The Inner product
+
* Orthonormal vectors
* Linear Combinations
+
* Orthogonal projections
 
+
* Orthogonal complements
 +
* Cauchy-Schwarz inequality
 +
* The angle between vectors
  
 
|-
 
|-
  
  
|Week&nbsp;3 
+
|
 +
  
||
+
|Week&nbsp;9
 
 
<div style="text-align: center;">2.1 and 2.2</div>
 
  
 
||
 
||
 
 
[[Linear Transformations]]
 
  
||
+
<div style="text-align: center;">6.3 and 6.4</div>
 
 
* [[Range of a Function]] <!-- 1073-Mod 1.2->
 
* [[Solutions of Linear Systems| Matrix addition]]  <!-- 2233-1.3--> 
 
* [[Transformations of Functions]]  <!-- 1073-Mod 6 -->
 
  
 
||
 
||
  
* Linear transformations and their properties
+
<div style="text-align: center;">5.2 and 5.3</div>
* Geometry of Linear Transformations (rotations, scalings and projections)
 
 
 
|-
 
 
 
 
 
|Week&nbsp;
 
  
 
||
 
||
 
+
       
<div style="text-align: center;"> 2.3 and 2.4</div>
+
<p> [[Orthonormal Bases and the Gram-Schmidt Process]] </p>
 
+
<p> [[Orthogonal Transformations and Orthogonal Matrices]] </p>
||
 
 
 
[[Matrix Products and Inverses]]  
 
  
 
||
 
||
  
* [[Solutions of Linear Systems| Linear Combinations]] <!-- 2233-1.3-->
+
* [[Subspaces of Rⁿ and Linear Independence]]  
* [[Inverse functions and the identity function|Inverse Functions]] <!-- 1073-7.2-->
+
* [[Dot Products and Orthogonality]]  
* [[Solutions of Linear Systems|Vectors and the Inner product]] <!-- 2233-1.3-->
 
  
 
||
 
||
  
* Matrix Products (both inner product and row-by-column methods)
+
* Orthogonal transformations
* The Inverses of a linear transform
+
* Orthonormal Bases
 +
* Orthogonal matrices
 +
* The transpose of a matrix
 +
* The Gram-Schmidt Process
 +
* QR factorization
  
 
+
|-   
|-
 
 
 
 
 
|Week&nbsp;6
 
 
 
||
 
 
 
<div style="text-align: center;">3.1</div>
 
 
 
||
 
 
 
[[Image and Kernel of a Linear Transform]]
 
 
 
||
 
 
 
* [[Solutions of Linear Systems]] <!-- 2233-1.3-->
 
* [[Range of a Function|Image of a Function]] <!-- 1073-Mod 1.2 -->
 
* [[Kernel of a Function]] <!-- DNE (recommend 1073 Mod 1.2 or Modern Algebra) -->
 
 
 
||
 
 
 
* The image of a Linear transformation
 
* The kernel of a linear transformation
 
* Span of a set of vectors
 
* Alternative characterizations of Invertible matrices
 
 
 
 
 
|-
 
 
 
 
 
 
 
|Week&nbsp;6
 
 
 
||
 
 
 
<div style="text-align: center;">3.2</div>
 
 
 
||
 
 
 
[[Linear Independence]]
 
 
 
||
 
 
   
 
   
* [[Solutions of Linear Systems]] <!-- 2233-1.3-->
 
* [[Image and Kernel of a Linear Transform]] <!-- 2233-3.1 -->
 
 
||
 
 
* Subspaces of R<sup>n</sup>
 
* Redundant vectors and linear independence
 
* Characterizations of Linear Independence
 
 
  
|-
+
|Week&nbsp;10
 
 
|Week&nbsp;6
 
  
 
||
 
||
  
<div style="text-align: center;">3.2</div>
+
<div style="text-align: center;">6.5 and 6.6</div>
  
 
||
 
||
 
 
[[Bases of Subspaces]]
 
  
||
+
<div style="text-align: center;">5.4</div>
 
* [[Linear Independence]] <!-- 2233-3.2-->
 
* [[Image and Kernel of a Linear Transform|The span of a set of vectors]] <!-- 2233-3.1 -->
 
  
 
||
 
||
 
+
       
* Bases and Linear independence
+
[[The Least-squares Solution]]
* Basis of the image
 
* Basis and unique representation
 
 
 
 
 
|-
 
 
 
 
 
|Week&nbsp;5
 
  
 
||
 
||
  
<div style="text-align: center;">3.3</div>
+
* [[Dot Products and Orthogonality]] 
 +
* [[The Column Space and Nullspace of a Linear Transformation]]
  
 
||
 
||
 
 
[[The Dimension of a Subspace]]
 
  
||
+
* 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
* [[Range of a Function|Image of a Function]]  <!-- 1073-Mod 1.2 -->
+
* Data fitting using the least-squares solution
* [[Bases and Linear Independence]] <!-- 2233-3.2 -->
 
* [[Linear transformations]] <!-- 2233-2.1-->
 
 
 
||
 
 
 
* Dimension of the Image
 
* Rank-nullity theorem
 
* Various bases in R<sup>n</sup>
 
  
  
Line 492: Line 384:
  
  
|Week&nbsp;7/8 
 
  
||
+
|Week&nbsp;11
 
 
<div style="text-align: center;"> 3.4  </div>
 
  
 
||
 
||
  
 
+
<div style="text-align: center;">3.1 and 3.2</div>
[[Similar Matrices and Coordinates]]
 
  
 
||
 
||
  
* [[Bases of Subspaces]] <!-- 2233-3.2 -->
+
<div style="text-align: center;">6.1 and 6.2</div>
* '''[[Equivalence Relations]]''' <!-- DNE (recommend 1073-Mod R) -->
 
  
 
||
 
||
 
+
       
* Coordinates in a subspace of R<sup>n</sup>
+
<p> [[Introduction to Determinants]] </p>
* Similar matrices
+
<p> [[Cramer's Rule]] </p>
* Diagonal matrices
 
  
 
||
 
||
  
 
+
* [[Orthonormal Bases and the Gram-Schmidt Process]] 
|-
+
* [[The Inverse of a Linear Transformation]]
 
+
* [[Sigma Notation]]
 
 
 
 
 
 
|Week&nbsp;9
 
 
 
||
 
 
 
<div style="text-align: center;"> 5.1</div>
 
  
 
||
 
||
 
 
[[Orthogonal Projections and Orthonormal Bases]]
 
  
||
+
* 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
  
* [[Parallel and Perpendicular Lines]] <!-- DNE (recommend 1093-2.1) -->
 
* [[Absolute value function]]<!-- DNE (recommend 1073-Mod R) -->
 
* [[Trig. Functions: Unit Circle Approach]] <!-- 1093-2.2 -->
 
* [[Matrix Products and Inverses|Inner Products]] <!-- 2233-2.3 and 2.4 -->
 
* [[Bases and Linear Independence]] <!-- 2233-3.2 -->
 
 
||
 
 
* 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&nbsp;10
+
|Week&nbsp;12
  
 
||
 
||
  
<div style="text-align: center;">5.2 </div>
+
<div style="text-align: center;">3.3</div>
  
 
||
 
||
 
 
[[Gram-Schmidt Process and QR Factorization]]
 
  
||
+
<div style="text-align: center;">6.3</div>
 
 
* [[Orthogonal Projections and Orthonormal Bases|Unit vectors]] <!-- 2233-5.1 and 5.2 -->  
 
* [[Matrix Products and Inverses|Inner Products]] <!-- 2233-2.3 and 2.4 -->
 
* [[Orthogonal Projections and Orthonormal Bases|Orthonormal Bases]] <!-- 2233-5.1 and 5.2 -->
 
* [[Bases and Linear Independence]]  <!-- 2233-3.2 -->  
 
  
 
||
 
||
 
+
       
* Gram-Schmidt process
+
[[The Geometric Interpretation of the Determinant]]
* QR Factorization
 
  
 
||
 
||
 
+
 
+
* [[Orthonormal Bases and the Gram-Schmidt Process]] 
|-
+
* [[Introduction to Determinants]]
 
+
* [[The Inverse of a Linear Transformation]]  
 
 
|Week&nbsp;11
 
 
 
||
 
 
 
<div style="text-align: center;">5.3</div>
 
 
 
||
 
 
 
[[Orthogonal Transformations and Orthogonal Matrices]]  
 
  
 
||
 
||
  
* [[Image and Kernel of a Linear Transform]] <!-- 2233-3.1 -->
 
* [[Matrix Products and Inverses|Inner Products]] <!-- 2233-2.3 and 2.4 -->
 
* [[Orthogonal Projections and Orthonormal Bases]]
 
  
||
+
* Cramer's Rule
 
+
* The adjoint and inverse of a matrix
* Orthogonal Transformations
+
* The area of a parallelogram and the volume of a parallelepiped
* Properties of Othogonal Transformations
 
* Transpose of a Matrix
 
* The matrix of an Orthogonal Projection
 
 
 
||
 
  
  
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|Week&nbsp;11
+
|Week&nbsp;13
  
 
||
 
||
  
<div style="text-align: center;">5.3</div>
+
<div style="text-align: center;">The beginning of 5.3 as well as the sections 5.1 and 5.2</div>
 
 
||
 
 
 
[[Least Squares]]
 
  
 
||
 
||
  
* [[Linear transformations]] <!-- 2233-2.1-->
+
<div style="text-align: center;">7.1, 7.2 and the beginning of 7.3</div>
* [[Orthogonal Transformations and Orthogonal Matrices]] <!-- 2233-5.3 -->
 
* [[Orthogonal Projections and Orthonormal Bases|Orthogonal Projections]] <!-- 2233-5.3 -->
 
  
 
||
 
||
 
+
       
* The Least Squares Solution
+
[[Eigenvalues and Eigenvectors]]
* The Normal Equation
 
* Another matrix for an Orthogonal Projection
 
  
 
||
 
||
  
 
+
* [[Introduction to Determinants]]   
|-
+
* [[The Column Space and Nullspace of a Linear Transformation]]
|Week&nbsp;11
+
* [[The Inverse of a Linear Transformation]]
  
 
||
 
||
  
<div style="text-align: center;">6.1 and 6.2</div>
+
* The requirement for a matrix to be diagonalizable
 
+
* Definition of an eigenvector
||
+
* The characteristic equation used to find eigenvalues
 
+
* Eigenvalues of a triangular matrix
[[Determinants]]
+
* Eigenspaces for specific eigenvalues
 
 
||
 
 
 
* [[Summation Notation]] <!-- DNE (recommend before Riemann Sums in 1214) -->
 
* [[Sgn Function]] <!-- DNE (recommend 1073 Mod R) -->
 
* [[Matrix Products and Inverses|Inverse of a Linear Transformation]] <!-- 2233-2.3 and 2.4 -->
 
* [[Orthogonal Transformations and Orthogonal Matrices| Transpose of a Matrix]] <!-- 2233-5.3 -->
 
 
 
||
 
 
 
* Properties of Determinants
 
* Sarrus's Rule
 
* Row operations and determinants
 
* Invertibility based on the determinant
 
 
 
||
 
  
  
 
|-
 
|-
  
|Week&nbsp;12
 
 
||
 
 
<div style="text-align: center;">6.3 </div>
 
  
||
+
|Week&nbsp;14
 
 
[[Cramer's Rule]]
 
  
 
||
 
||
  
* [[Determinants]] <!-- 2233- 5.3 -->
+
<div style="text-align: center;">5.3 and 5.4</div>
* [[Matrix Products and Inverses| Invertible matrices]] <!-- 2233- 2.3 and 2.4 -->
 
* [[Linear Transformations| Rotations]] <!-- 2233- 2.1 and 2,2 -->  
 
  
 
||
 
||
  
* Parrallelepipeds in R<big>n</big>
+
<div style="text-align: center;">3.4 and 7.3</div>
* Geometric Interpretation of the Determinant
 
* Cramer's rule
 
  
 
||
 
||
 
+
       
 
+
[[Diagonalization of Matrices]]
|-
 
 
 
 
 
|Week&nbsp;13
 
  
 
||
 
||
  
<div style="text-align: center;">7.1</div>
+
* [[Eigenvalues and Eigenvectors]] 
 +
* [[The Column Space and Nullspace of a Linear Transformation]]
  
 
||
 
||
 
 
[[Diagonalization]]
 
  
||
+
* Similar matrices
 
+
* Diagonalization in terms of linearly independent eigenvectors
* [[Similar Matrices and Coordinates]] <!-- 2233- 3.4 -->
+
* Algebraic and geometric multiplicity for a specific eigenvalue
* [[Orthogonal Transformations and Orthogonal Matrices]] <!-- 2233- 5.3 -->
+
* The strategy for diagonalization
 
 
||
 
 
 
* Diagonalizable matrices
 
* Eigenvalues and eigenvectors
 
* Real eigenvalues of orthogonal matrices
 
 
 
||
 
 
 
 
 
|-
 
 
 
|Week&nbsp;14
 
 
 
||
 
 
 
<div style="text-align: center;">7.2 and 7.3</div>
 
 
 
||
 
 
 
[[Finding Eigenvalues and Eigenvectors]]
 
 
 
||
 
 
 
* [[Determinants]] <!-- 2233- 5.3 -->
 
* [[Matrix Products and Inverses| Invertible matrices]] <!-- 2233- 2.3 and 2.4 -->
 
* [[Diagonalization]] <!-- 2233- 7.1 -->
 
* [[Image and Kernel of a Linear Transform]] <!-- 2233- 3.1 -->
 
 
 
||
 
 
 
* Eigenvalues from the characteristic equation
 
* Eigenvalues of Triangular matrices
 
* Characteristic Polynomial
 
* Eigenspaces and eigenvectors
 
* Geometric and algebraic multiplicity
 
* Eigenvalues of similar matrices
 
 
 
 
 
|-
 
 
 
|Week&nbsp;14
 
 
 
||
 
 
 
<div style="text-align: center;">8.1</div>
 
 
 
||
 
 
 
[[Symmetric Matrices]]
 
 
 
||
 
  
* [[Similar Matrices and Coordinates]] <!-- 2233- 8.1 -->
 
* [[Orthogonal Transformations and Orthogonal Matrices| Transpose of a Matrix]] <!-- 2233- 5.3  -->
 
* [[Diagonalization|Eigenvalues and Eigenvectors]] <!-- 2233- 7.1 -->
 
* [[Finding Eigenvalues and Eigenvectors|Algebraic and Geometric Multiplicities]] <!-- 2233- 7.2 and 7.3 -->
 
  
||
 
 
* Orthogonally Diagonalizable Matrices
 
* Spectral Theorem
 
* The real eigenvalues of a symmetric matrix
 
  
 
|-
 
|-
 
|Week&nbsp;14
 
 
||
 
 
<div style="text-align: center;">8.2</div>
 
 
||
 
 
 
[[Quadratic Forms]]
 
 
||
 
 
* [[Symmetric Matrices]] <!-- 2233- 8.1 -->
 
* [[Finding Eigenvalues and Eigenvectors]] <!-- 2233- 7.2 and 7.3 -->
 
* '''[[Conics]]''' <!-- DNE (recommend 1093 or do not include discussion on Principal axes in this topic -->
 
 
||
 
 
* Quadratic Forms
 
* Diagonalizing a Quadratic Form
 
* Definiteness of a Quadratic Form
 
* '''Principal Axes''' <!-- May not include if conics are not discussed prior  -->
 
* '''Ellipses and Hyperbolas from Quadratic Forms'''  <!-- May not include if conics are not discussed prior  -->
 
 
||
 

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