The course website for Spring 2018 is here
Instructor: Yu Tsumura
Where: Macquigg Laboratory 160
When: MWF 1:502:45
Textbook
The required textbook is
Introduction to Linear Algebra, 5th edition,
by L.W. Johnson, R.D. Riess, and J.T. Arnold, published by Pearson,
ISBN Softcover: 0321628217, Hardcover: 0201658593
Lecture Notes
 Lecture Notes 1 §1.1 Introduction to Matrices and Systems of Linear Equations
 Lecture Notes 2 §1.2 Echelon Form and GaussJordan Elimination
 Lecture Notes 3 §1.3 Consistent Systems of Linear Equations
 Lecture Notes 4 §1.5 Matrix Operations
 Lecture Notes 5 §1.6 Algebraic Properties of Matrix Operations
 Lecture Notes 6 §1.7 Linear Independence and Nonsingular Matrices
 Lecture Notes 7 §1.7 Linear Independence and Nonsingular Matrices Part 2
 Lecture Notes 8 §1.9 Matrix Inverse and Their Properties
 Lecture Notes 9 §1.9 Matrix Inverse and Their Properties Part 2
 Lecture Notes 10 §3.2 Vector Space Properties of $\R^n$
 Lecture Notes 11 §3.3 Examples of Subspaces
 Lecture Notes 12 §3.3 Examples of Subspaces Part 2
 Midterm 1. Bring the Buck ID with you.
 Lecture 14. Review of Midterm 1.
 Lecture Notes 15 §3.4 Bases for Subspaces
 Lecture Notes 16 §3.4 Bases for Subspaces Part 2
 Lecture Notes 17 §3.4 Part 3 and §3.5 Dimensions
 Lecture Notes 18 §3.5 Dimensions Part 2
 Lecture Notes 19 §5.2 Vector Spaces
 Lecture Notes 20 §5.2 Part 2 & §5.3 Subspaces
 Lecture Notes 21 §5.3 Subspaces Part 2
 Lecture Notes 22 §5.4 Linear Independence, Bases, and Coordinates
 Lecture Notes 23 §5.4 Linear Independence, Bases, and Coordinates Part 2
 Lecture Notes 24 §5.4 part 3 & §3.6 Orthogonal Bases for Subspaces
 Lecture Notes 25 §3.6 Orthogonal Bases for Subspaces Part 2
 Lecture Notes 26 §3.7 Linear Transformation from $\R^n$ to $\R^m$.
 Lecture Notes 27 §3.7 Part 2 $\R^n$ to $\R^m$.
 Lecture Notes 28 §4.1 The Eigenvalue Problem for $(2\times 2)$ Matrices
 Lecture Notes 29: In class review session for midterm 2
 Midterm 2
 Lecture Notes 31 §4.2 Determinants and the Eigenvalue Problem
 Lecture 32 Returned and explained the solutions of Midterm 2.
 Lecture Notes 33 §4.2 Part 2 and §4.4 Eigenvalues and the Characteristic Polynomial
 Lecture Notes 34 §4.4 Eigenvalues and the Characteristic Polynomial Part 2
 Lecture Notes 35 §4.5 Eigenvectors and Eigenspaces
 Lecture Notes 36 4.5 Part 2 and §4.6 Complex Eigenvalues and Eigenvectors
 Lecture Notes 37 §4.6 Part 2 and §4.7 Similarity Transformation and Diagonalization
 Lecture Notes 38 §4.7 Similarity Transformation and Diagonalization part 2
 Lecture Notes 39 §4.7 Similarity Transformation and Diagonalization part 3
 Practice for the final exam 1
 Practice for the final exam 2
Midterm 1 Information
We will have midterm 1 on 9/22 Friday in class.
The exam will cover the materials we studied in Chapter 1 of the textbook.
(Section 1.4 and 1.8 are excluded.)
Please review lecture notes, homework problems (including extra problems).
There are supplementary/conceptual exercises in the textbook starting on page 105.
Midterm 1 Problems and Solutions.
More practice problems for midterm 1
Check out the list of linear algebra problems and study problems from Chapter 1.
The followings are past exam problems from Spring 2017.
 Problem 1 and its solution: Possibilities for the solution set of a system of linear equations
 Problem 2 and its solution: The vector form of the general solution of a system
 Problem 3 and its solution: Matrix operations (transpose and inverse matrices)
 Problem 4 and its solution: Linear combination
 Problem 5 and its solution: Inverse matrix
 Problem 6 and its solution: Nonsingular matrix satisfying a relation
 Problem 7 and its solution: Solve a system by the inverse matrix
 Problem 8 and its solution:A proof problem about nonsingular matrix
Midterm 2 Information
We will have midterm 2 on 11/3 Friday in class.
The exam will cover the materials we studied in Chapter 3 and 5 of the textbook.
Midterm 2 Problems and Solutions
 Vector Space of 2 by 2 Traceless Matrices
 Find an Orthonormal Basis of the Given Two Dimensional Vector Space
 Are the Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Linearly Independent?
 Find Bases for the Null Space, Range, and the Row Space of a $5\times 4$ Matrix
 Matrix Representation, Rank, and Nullity of a Linear Transformation $T:\R^2\to \R^3$
 Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors
 Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less
More practice problems for midterm 2
Check out the list of linear algebra problems and study problems from Chapter 3 and 5.
The followings are past exam problems from Spring 2017.
 True of False Problems and Solutions: True or False problems of vector spaces and linear transformations
 Problem 1 and its solution: See (7) in the post "10 examples of subsets that are not subspaces of vector spaces"
 Problem 2 and its solution: Determine whether trigonometry functions $\sin^2(x), \cos^2(x), 1$ are linearly independent or dependent
 Problem 3 and its solution: Orthonormal basis of null space and row space
 Problem 4 and its solution: Basis of span in vector space of polynomials of degree 2 or less
 Problem 5 and its solution: Determine value of linear transformation from $\R^3$ to $\R^2$
 Problem 6 and its solution: Rank and nullity of linear transformation from $\R^3$ to $\R^2$
 Problem 7 and its solution: Find matrix representation of linear transformation from $\R^2$ to $\R^2$
 Problem 8 and its solution: Hyperplane through origin is subspace of 4dimensional vector space
Homework Assignments
Please use the cover sheet for the homework assignments.
(You may reuse the cover sheet. Please remove the old staple neatly and staple again.)
HW 1 (Due 8/30 in class)
Do and submit your solutions of the following problems from the textbook.
 Lecture 1 §1.1 p.12 #2, 20, 28.
Also take a quiz here. (No need to submit this quiz. This is just for your practice. You can take the quiz as many times as you want.)  Lecture 2 §1.2 p.26 #18, 28, 44.
Note that for #44, "the matrix $A$ is row equivalent to the matrix $I$" means that if you apply several elementary row operations, then $A$ is transformed into $I$.
For those who has not received the textbook yet, pictures of the problems are available here (only this time).
Extra (Not submit these)
 Lecture 1 §1.1 #3, 4, 11, 24, 32
 Lecture 2 §1.2 #3, 9, 15, 17, 23, 37, 54
Solution 1
HW 2 (Due 9/6 in class)
Do and submit your solutions of the following problems from the textbook.
 Lecture 3 §1.3 p.38 #4, 14, 23.
Also take a quiz and read Summary: Possibilities for the Solution Set of a System of Linear Equations.  Lecture 4 §1.5 p.57 # 48, 58, 68
 Lecture 5 §1.6 p.69 #26, 30, 43
Please use the cover sheet (can be found above).
(You may reuse the cover sheet from HW 1.)
Extra (Not submit these)
 Lecture 3 §1.3 p.38 #1, 8, 10, 12, 16, 20, 24.
 Lecture 4 §1.5. p.57 #1, 2, 3, 6, 8, 12, 25, 26, 30, 31, 32, 36, 38, 42, 53, 54, 55, 56, 65, 66
 Lecture 5 §1.6. p.69 #13, 15, 27, 28, 31, 34, 42, 46, 49
HW 3 (Due 9/13 in class)
Do and submit your solutions of the following problems from the textbook.
 Lecture 6 §1.7 p.78. #12, 38, 51 and the following problem.
Let
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
1 \\
a \\
5
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
0 \\
4 \\
b
\end{bmatrix}\] be vectors in $\R^3$.
Determine a condition on the scalars $a, b$ so that the set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is linearly dependent.  Lecture 7 §1.7 p.78.#23, 44, 46, 48, 56
Extra (Not submit these)
 Lecture 6 §1.7 p.78. 4, 11, 13, 30, 31, 39, 46, 49, 50, 52
 Lecture 7 §1.7 p.78. 40, 54, 55, 57
Solution 3
The solution of the above problem is given ↴
Determine a Condition on $a, b$ so that Vectors are Linearly Dependent
For Ex 46 (a), A geometric picture of this problem is that in a plane, any three vectors must be linearly dependent. The explanation is the following: the linear dependence relation of the three vectors is equivalent to solving a homogeneous linear system with 3 variables, and 2 equations, which means, the system always admits a nonzero solution, and this proves that three vectors are linearly dependent.
HW 4 (Due 9/20 in class)
Do and submit your solutions of the following problems from the textbook.
Extra (Not submit these)
 Lecture 8 §1.9 p.102 #1, 9, 13, 23, 27, 29, 30
 Lecture 9 §1.9 p.102 #39, 40, 51, 52, 54, 55, 56, 72, 74
Solution 4
Graded problems: Ex28 (3 points), Ex 50(4 points), and Ex 67(3 points).
For Ex 28, to find $\lambda$ such that the matrix is invertible is the same as the matrix is nonsingular. As what as we did before, we do row reduction to make the matrix into echelon form (not need to be reduced echelon form) and such that the first and second rows contain only numbers, while the 3rd row is $(0,0,\lambda34/7)$. Now, the matrix is invertible is the same as saying $\lambda34/7$ not equal to $0$. Details see the answer.
For Ex 50, from the equation $A^2=AB+2A$, multiplying the $A^{1}$ on the left, we get $A^{1}A^2= A^{1}AB+ 2A^{1}A$, this is $A=B+2I$, but if you multiply $A^{1}$ on the right, you will get: $A=ABA^{1}+2I$. Recall by an exercise in HW2, we know in general $AB$ is not equal to $BA$, and so in general $ABA^{1}$ is not equal to $B$, and this equation is not what we want.
HW 5 (Due 9/27 in class)
Do and submit your solutions of the following problems from the textbook.
Extra (Not submit these)
 Lecture 11 §3.2 p.174 # 2, 3, 11, 12, 13, 16, 19, 30, 31
 Lecture 12 §3.3 p.186 #4, 6, 9, 15, 17, 20, 22, 25, 26, 34, 40, 43, 46, 49
Solution 5
Graded problems: problem 8 of section 3.2 and problem 30, 38 in section 3.3.
For Ex 8, verify that both $(0,1)$ and $(1,0)$ are in the set $W$, but their addition, $(1,1)$ is not in $W$.
For Ex 30, when you want to describe the range of a matrix $\calR(A)$ algebraically, you should try to write it as linear combinations of some vectors. For example, in this problem, it is a linear combination of $[0,1]$ and $[1,0]$, or it is the whole space $\R^2$.
HW 6 (Due 10/4 in class)
Do and submit your solutions of the following problems from the textbook.
(You may only use the definition of a nonsingular matrix.)
1 & 1 & 1 \\
0 &1 &0 \\
2 & 1 & 4
\end{bmatrix}.\] Find a nonsingular matrix $A$ satisfying
\[A^2=AB+2A\] if exists. If you think such a nonsingular matrix does not exist, exlain why.
\[A=BC,\] where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix.
Prove that the matrix $A$ cannot be invertible.
Extra (Not submit these)
 Lecture 14 Review Midterm 1 Problems and Solutions.
 Lecture 15 §3.4 p.200 #6, 9, 10, 27, 33, 34, 36
 Lecture 15 §3.4 p.200 #11, 14, 17, 22, 30, 31
Solution 6
The solutions of Problem A, Problem B, and Problem C.
For Ex 24, many of you are wrong in part (a). To be more precise, many of you find the basis for $\Span(S)$ containing 2 vectors, and in part (b) find another basis containing 3 vectors. This is a contradiction, since number of vectors in a basis for a space should be the same. Please see solution to see the details.
HW 7 (Due 10/11 in class)
Do and submit your solutions of the following problems from the textbook.
Extra (Not submit these)
 Lecture 17 §3.5 p. 212 #4, 6, 8, 12, 18, 28,
 Lecture 18 §3.5 p. 212 #35, 37, 38, 39
Solution 7
Note that if you get points off, you may fail to explain why the system admits a nonzero solution, so you may not start from the equation $c_{1}A_{1}+c_{2}A_{2}+c_{3}A_{3}+A_{4}=\mathbf{0}$.
HW 8 (Due 10/18 in class)
Do and submit your solutions of the following problems from the textbook.
Extra (Not submit these)
 Lecture 20 §5.2 p.366 #10, 11, 13, 16, 17, 21, 26
 Lecture 21 §5.3 p.373 3, 7, 12, 13, 17, 19, 23, 27
Solution 8
For Ex 4 in Section 5.3, many of you know that $W$ is not a subspace of $V$, in particular, $W$ doesn't satisfy (S2). However, to prove $W$ fails in (S2), you should prove there is some $A$ and $B$ in $W$, but $A+B$ is not in $W$, that is, you should provide a counterexample. Please see the solution for details.
HW 9 (Due 10/25 in class)
Do and submit your solutions of the following problems from the textbook.
0 & 0\\
2& 0
\end{bmatrix}$)
Extra (Not submit these)
 Lecture 22, 23, 24 §5.4 p.387 #1, 5, 9, 10, 11, 12, 14, 15, 18, 27, 36, 37
Solution 9
HW 10 (Due 11/1 in class)
Do and submit your solutions of the following problems from the textbook.
Extra (Not submit these)
 Lecture 25 §3.6 p.224 #1, 5, 12, 13, 21, 28
 Lecture 26, 27 §3.7 p.239 #1, 2, 3, 4, 6, 13, 18, 20, 21, 25, 30
Solution 10
1 \\
1
\end{bmatrix}$ and $\begin{bmatrix}
1 \\
1
\end{bmatrix}$, so we need to express the standard basis vectors as linear combinations of vectors in the new basis, and then use the linearity property of $T$ (i.e., the equation (6)). Please see the solution for details. "
HW 11 (Due 11/8 in class)
Do and submit your solutions of the following problems from the textbook.
Extra (Not submit these)
 Lecture 28 §4.1 p.279 #1, 3, 7, 13, 18, 19
Solution 11
0 & 1\\
1& 0
\end{bmatrix}$ does not have a real eigenvalue). Showing there is an eigenvalue is the same as showing there is a root for the quadric $\lambda^2(a+d)\lambda+(adb^2)$. But the discriminant of the quadric is $(ad)^2+4b^2$, which is greater or equal to zero, meaning the quadric has at least one root. This is what we want to show. Please refer to solution for a complete proof.
HW 12 (Due 11/15 in class)
Do and submit your solutions of the following problems from the textbook.
 Lecture 31 §4.2 p.288 #14, 18, 21, 29
 Lecture 32
Problem A. Let $T:\R^2\to \R^2$ be a linear transformation such that it maps the vectors $\mathbf{v}_1, \mathbf{v}_2$ as indicated in the figure below.
Find the matrix representation $A$ of the linear transformation $T$.
Problem B. An $n\times n$ matrix $A$ is called orthogonal if $A^{\trans}A=I$.
Let $V$ be the vector space of all real $2\times 2$ matrices.
Consider the subset
\[W:=\{A\in V \mid \text{$A$ is an orthogonal matrix}\}.\] Prove or disprove that $W$ is a subspace of $V$.Problem C. Let $C[2\pi, 2\pi]$ be the vector space of all realvalued continuous functions defined on the interval $[2\pi, 2\pi]$.
Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$.
(a) Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ is a basis for $W$.(b) Prove that the set $\{\sin^2(x)\cos^2(x), 1\}$ is a basis for $W$.
(Hint: Use the coordinate vectors with respect to the basis $B$ from part (a).)
Extra (Not submit these)
 Lecture 31 §4.2 p.288 #17, 19, 22, 24, 26, 30
 Lecture 32 Review and redo midterm 2 problems. Compare your solutions with mine.
Solution 12
Solutions to Problem A, Problem B, Problem C
HW 13 (Due 11/20 Monday in class)
Do and submit your solutions to the following problems from the textbook.
 Lecture 33 §4.4 p.305 #3, 8, 14
 Lecture 34 §4.4 p.305 #9, 18, 19
 Lecture 35 §4.5 p.314 #12, 16, 18
Extra (Not submit these)

 Lecture 33 §4.4 p.305 #1, 9, 11, 15, 16
 Lecture 34 §4.4 p.305 #12, 17, 21, 25
 Lecture 35 §4.5 p.314 #3, 9, 11, 17, 19, 21, 22
Solution 13
0 \\
9
\end{bmatrix}$, which is hard to compute directly, but we know the same expression is easy to compute when $X$ is an eigenvector of $A$. After finding the eigenvectors $\begin{bmatrix}
1 \\
1
\end{bmatrix}, \begin{bmatrix}
2 \\
5
\end{bmatrix}$ of $A$, we simply need to express $\begin{bmatrix}
0 \\
9
\end{bmatrix}=a\begin{bmatrix}
1 \\
1
\end{bmatrix}+b\begin{bmatrix}
2 \\
5
\end{bmatrix}$ and find coefficients $a, b$, and everything will follow.
HW 14 (Due 11/29 Wednesday in class)
Do and submit your solutions to the following problems from the textbook.
 Lecture 36 §4.5 p.314 #17, 22 and §4.6 p.324 #21, 23, 38
Extra (Not submit these)

 Lecture 36 §4.6 p.324 #16, 19, 22, 29, 39, 40, 41, 42
 Please take the textbook survery.
HW (No submit)
The following problems will not be collected and graded. However, you need to do these problems to prepare for the final exam.
 Lecture 37 §4.7 p.314 #25 and do this and this.
 Lecture 38 §4.7 p.314 #1, 3, 4, 9, 10, 11
 Lecture 39 If you haven't done, please take the survery about the textbook.
Linear Algebra Q&A
The Q&A broad for students taking Math 2568 at OSU.
You can ask and answer questions regarding Math 2568 linear algebra.
Help each other.