Math 220: Linear Algebra
Spring 2011
Procedural Documents [Top Contents]
- Syllabus (Word: Syllabus)
- Calendar (Excel: Calendar and Homework)
- Homework (Excel: Calendar and Homework)
- Projects: one (rubric), two (rubric), and three (rubric)
About Linear Algebra
- A brief history of linear algebra (here)
- Pronunciation guide (here)
- Proof: The Language of Mathematics (here)
- Seeing the Light (A PowerPoint on using Linear Algebra for lighting graphics in 3D)
Topics of Study [Top Contents]
| Topic | Handouts | Lectures Notes | |
| Linear Equations 1.1: Intro to Linear Systems 1.2: Matrices, Vectors, and Gauss-Jordan Elimination 1.3: Matrix Algebra |
1.1 text 1.2 text 1.3 text |
1.1 notes 1.2 notes 1.3 notes |
|
| Linear Transformations 2.1: Intro to Linear Transformations and Their Inverses 2.2: Linear Transformations in Geometry 2.3: Matrix Products 2.4: The Inverse of a Linear Transformation |
2.1 text (Example) 2.2 text 2.3 text 2.4 text |
2.1 notes 2.2 notes 2.3 notes 2.4 notes |
|
| Subspaces of Rn and Their
Dimension 3.1: Image and Kernel 3.2: Subspaces; Bases and LI 3.3: The Dimension of a Subspace 3.4: Coordinates |
3.1 notes 3.2 notes 3.3 notes 3.4 notes |
||
| Orthogonality and Least Squares 5.1: Orthogonal Projections and Bases 5.2: Gram-Schmidt and QR Factorization 5.3: Orthogonal Transformations and Matrices 5.4: Least Squares and Data Fitting |
5.1 notes 5.2 notes 5.3 notes 5.4 notes |
||
| Determinants 6.1: Intro to Determinants 6.2: Properties of Determinants 6.3: Geometrical Interpretations of the Determinant |
6.1 notes 6.2 notes 6.3 notes |
||
| Eigenvalues and Eigenvectors 7.1: An Introductory Example 7.2: Finding the Eigenvalues of a Matrix 7.3: Finding the Eigenvectors of a Matrix 7.4: Diagonalization 7.5: Complex Eigenvalues 7.6: Stability |
7.1 notes 7.2 notes 7.3 notes 7.4 notes 7.5 notes 7.6 notes |
Study Guides, Tests and Test Keys [Top Contents]
| Topic | Review | Previous Test (Key) | |
| Test 1 | T/F Chapter Review (1 and 2) | Test and Key | Test (Key) |
| Test 2 | Test and Key | Test (Key) | |
| Test 3 | Test and Key | Test (Key) | |
| Final Exam |
Videos by Topic (from the Khan Academy)
2.
Matrix multiplication (part 1)
3.
Matrix multiplication (part 2)
5.
Inverting matrices (part 2)
6.
Inverting Matrices (part 3)
7.
Matrices to solve a system of
equations
8.
Matrices to solve a vector combination
problem
10.
3-variable linear equations (part 1)
11.
Solving 3 Equations with 3 Unknowns
12.
Linear Algebra: Introduction to
Vectors
13.
Linear Algebra: Vector Examples
14.
Linear Algebra: Parametric
Representations of Lines
15.
Linear Combinations and Span
16.
Linear Algebra: Introduction to Linear
Independence
17.
More on linear independence
18.
Span and Linear Independence Example
19.
Linear Subspaces
20.
Linear Algebra: Basis of a Subspace
21.
Vector Dot Product and Vector Length
22.
Proving Vector Dot Product Properties
23.
Proof of the Cauchy-Schwarz Inequality
24.
Linear Algebra: Vector Triangle
Inequality
25.
Defining the angle between vectors
26.
Defining a plane in R3 with a point
and normal vector
27.
Linear Algebra: Cross Product
Introduction
28.
Proof: Relationship between cross
product and sin of angle
29.
Dot and Cross Product
Comparison/Intuition
30.
Matrices: Reduced Row Echelon Form 1
31.
Matrices: Reduced Row Echelon Form 2
32.
Matrices: Reduced Row Echelon Form 3
34.
Introduction to the Null Space of a
Matrix
35.
Null Space 2: Calculating the null
space of a matrix
36.
Null Space 3: Relation to Linear
Independence
38.
Null Space and Column Space Basis
39.
Visualizing a Column Space as a Plane
in R3
40.
Proof: Any subspace basis has same
number of elements
41.
Dimension of the Null Space or Nullity
42.
Dimension of the Column Space or Rank
43.
Showing relation between basis cols
and pivot cols
44.
Showing that the candidate basis does
span C(A)
45.
A more formal understanding of
functions
48.
Matrix Vector Products as Linear
Transformations
49.
Linear Transformations as Matrix
Vector Products
50.
Image of a subset under a
transformation
51.
im(T): Image of a Transformation
53.
Preimage and Kernel Example
54.
Sums and Scalar Multiples of Linear
Transformations
55.
More on Matrix Addition and Scalar
Multiplication
56.
Linear Transformation Examples:
Scaling and Reflections
57.
Linear Transformation Examples:
Rotations in R2
58.
Rotation in R3 around the X-axis
59.
Unit Vectors
60.
Introduction to Projections
61.
Expressing a Projection on to a line
as a Matrix Vector prod
62.
Compositions of Linear Transformations
1
63.
Compositions of Linear Transformations
2
64.
Linear Algebra: Matrix Product
Examples
65.
Matrix Product Associativity
66.
Distributive Property of Matrix
Products
67.
Linear Algebra: Introduction to the
inverse of a function
68.
Proof: Invertibility implies a unique
solution to f(x)=y
69.
Surjective (onto) and Injective
(one-to-one) functions
70.
Relating invertibility to being onto
and one-to-one
71.
Determining whether a transformation
is onto
72.
Exploring the solution
set of Ax=b
73.
Matrix condition for
one-to-one trans
74.
Simplifying conditions
for invertibility
75.
Showing that Inverses
are Linear
76.Deriving a method for
determining inverses
77.
Example of Finding
Matrix Inverse
79.
3x3 Determinant
80.
nxn Determinant
81.
Determinants along
other rows/cols
82.
Rule of Sarrus of
Determinants
83.
Determinant when row
multiplied by scalar
84.
(correction) scalar
muliplication of row
85.
Determinant when row
is added
87. Determinant after row
operations
88.Upper Triangular
Determinant
90.
Determinant and area
of a parallelogram
91.
Determinant as Scaling
Factor
94.
Transposes of sums and
inverses
96.Rowspace and Left
Nullspace
97.
Visualizations of Left
Nullspace and Rowspace
99.
Rank(A) =
Rank(transpose of A)
100. dim(V) +
dim(orthogonoal complelent of V)=n
101.Representing vectors in Rn
using subspace members
102. Orthogonal Complement of the
Orthogonal Complement
103.Orthogonal Complement of the
Nullspace
104.Unique rowspace solution to
Ax=b
105.Rowspace Solution to Ax=b
example
106.Showing that A-transpose x A
is invertible
107.Projections onto
Subspaces
108.Visualizing a projection
onto a plane
109.A Projection onto a Subspace
is a Linear Transforma
110.Subspace Projection
Matrix Example
111.Another Example of a
Projection Matrix
112.Projection is closest
vector in subspace
113. Least Squares
Approximation
115.Another Least Squares
Example
116.Linear Algebra: Coordinates with
Respect to a Basis
118.Invertible Change of Basis
Matrix
119. Transformation Matrix with
Respect to a Basis
120. Alternate Basis Tranformation
Matrix Example
121.Alternate Basis Tranformation
Matrix Example Part 2
122. Changing coordinate
systems to help find a transformation matrix
123. Introduction to
Orthonormal Bases
124. Coordinates with
respect to orthonormal bases
125. Projections onto subspaces
with orthonormal bases
126.Finding projection onto subspace
with orthonormal basis example
127. Example using
orthogonal change-of-basis matrix to find transformation matrix
128. Orthogonal matrices preserve
angles and lengths
130. Gram-Schmidt Process
Example
131.Gram-Schmidt example
with 3 basis vectors
132. Introduction to
Eigenvalues and Eigenvectors
133. Proof of formula for
determining Eigenvalues
134
Example solving for the
eigenvalues of a 2x2 matrix
135.Finding Eigenvectors
and Eigenspaces example
136. Eigenvalues of a 3x3
matrix
137.Eigenvectors and
Eigenspaces for a 3x3 matrix
138.Showing that an
eigenbasis makes for good coordinate systems
139.Vector Triple Product Expansion (very
optional)
140. Normal vector from plane equation
141.Point distance to plane
Distance Between Planes
Video Lectures from MIT: Additional supplemental materials (including these video lectures) may be found on the MIT OpenCourseWare site