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by Yu · Published 10/15/2017
Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix
Consider the Hermitian matrix
\[A=\begin{bmatrix}
-i& 1
\end{bmatrix}.\]
(a) Find the eigenvalues of $A$.
(b) For each eigenvalue of $A$, find the eigenvectors.
(c) Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix $D$ and a unitary matrix $U$ such that $U^{-1}AU=D$.
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by Yu · Published 10/12/2017 · Last modified 10/13/2017
A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix
Prove that the matrix
-1& 0
\end{bmatrix}\] is diagonalizable.
Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix.
That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal matrix.
Diagonalize the Upper Triangular Matrix and Find the Power of the Matrix
Consider the $2\times 2$ complex matrix
a & b-a\\
0& b
(b) For each eigenvalue of $A$, determine the eigenvectors.
(c) Diagonalize the matrix $A$.
(d) Using the result of the diagonalization, compute and simplify $A^k$ for each positive integer $k$.
Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible?
A square matrix $A$ is called nilpotent if some power of $A$ is the zero matrix.
Namely, $A$ is nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix.
Suppose that $A$ is a nilpotent matrix and let $B$ be an invertible matrix of the same size as $A$.
Is the matrix $B-A$ invertible? If so prove it. Otherwise, give a counterexample.
The Subspace of Linear Combinations whose Sums of Coefficients are zero
Let $V$ be a vector space over a scalar field $K$.
Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset
\[W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } a_1+a_2+\cdots+a_k=0\}.\] So each element of $W$ is a linear combination of vectors $\mathbf{v}_1, \dots, \mathbf{v}_k$ such that the sum of the coefficients is zero.
Prove that $W$ is a subspace of $V$.
The Column Vectors of Every $3\times 5$ Matrix Are Linearly Dependent
(a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly dependent.
(b) Prove that the row vectors of every $5\times 3$ matrix $B$ are linearly dependent.
Determine Whether Each Set is a Basis for $\R^3$
Determine whether each of the following sets is a basis for $\R^3$.
(a) $S=\left\{\, \begin{bmatrix}
1 \\
\end{bmatrix}, \begin{bmatrix}
-2 \\
\end{bmatrix} \,\right\}$
(b) $S=\left\{\, \begin{bmatrix}
(c) $S=\left\{\, \begin{bmatrix}
(d) $S=\left\{\, \begin{bmatrix}
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Find the Dimension of the Subspace of Vectors Perpendicular to Given Vectors
Let $V$ be a subset of $\R^4$ consisting of vectors that are perpendicular to vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, where
\[\mathbf{a}=\begin{bmatrix}
\end{bmatrix}, \quad \mathbf{b}=\begin{bmatrix}
\end{bmatrix}, \quad \mathbf{c}=\begin{bmatrix}
Namely,
\[V=\{\mathbf{x}\in \R^4 \mid \mathbf{a}^{\trans}\mathbf{x}=0, \mathbf{b}^{\trans}\mathbf{x}=0, \text{ and } \mathbf{c}^{\trans}\mathbf{x}=0\}.\]
(a) Prove that $V$ is a subspace of $\R^4$.
(b) Find a basis of $V$.
(c) Determine the dimension of $V$.
Every Basis of a Subspace Has the Same Number of Vectors
Let $V$ be a subspace of $\R^n$.
Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ is a basis of the subspace $V$.
Prove that every basis of $V$ consists of $k$ vectors in $V$.
If there are More Vectors Than a Spanning Set, then Vectors are Linearly Dependent
Suppose that
\[S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m\}\] is a spanning set for $V$.
Prove that any set of $m+1$ or more vectors in $V$ is linearly dependent.
If a Half of a Group are Elements of Order 2, then the Rest form an Abelian Normal Subgroup of Odd Order
Let $G$ be a finite group of order $2n$.
Suppose that exactly a half of $G$ consists of elements of order $2$ and the rest forms a subgroup.
Namely, suppose that $G=S\sqcup H$, where $S$ is the set of all elements of order in $G$, and $H$ is a subgroup of $G$. The cardinalities of $S$ and $H$ are both $n$.
Then prove that $H$ is an abelian normal subgroup of odd order.
Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis.
Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$.
(a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$.
(b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of $\R^3$.
Polynomial Ring with Integer Coefficients and the Prime Ideal $I=\{f(x) \in \Z[x] \mid f(-2)=0\}$
Let $\Z[x]$ be the ring of polynomials with integer coefficients.
Prove that
\[I=\{f(x)\in \Z[x] \mid f(-2)=0\}\] is a prime ideal of $\Z[x]$. Is $I$ a maximal ideal of $\Z[x]$?
Linear Algebra Midterm 1 at the Ohio State University (3/3)
The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.
There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).
The time limit was 55 minutes.
This post is Part 3 and contains Problem 7, 8, and 9.
Check out Part 1 and Part 2 for the rest of the exam problems.
Problem 7. Let $A=\begin{bmatrix}
8& 9
\end{bmatrix}$ and $\mathbf{v}=\begin{bmatrix}
\end{bmatrix}$.
(a) Calculate $A\mathbf{v}$ and find the number $\lambda$ such that $A\mathbf{v}=\lambda \mathbf{v}$.
(b) Without forming $A^3$, calculate the vector $A^3\mathbf{v}$.
Problem 8. Prove that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular.
Problem 9.
Determine whether each of the following sentences is true or false.
(a) There is a $3\times 3$ homogeneous system that has exactly three solutions.
(b) If $A$ and $B$ are $n\times n$ symmetric matrices, then the sum $A+B$ is also symmetric.
(c) If $n$-dimensional vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are linearly dependent, then the vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4$ is also linearly dependent for any $n$-dimensional vector $\mathbf{v}_4$.
(d) If the coefficient matrix of a system of linear equations is singular, then the system is inconsistent.
(e) The vectors
\[\mathbf{v}_1=\begin{bmatrix}
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
\end{bmatrix}\] are linearly independent.
Problem 4. Let
\[\mathbf{a}_1=\begin{bmatrix}
\end{bmatrix}, \mathbf{a}_2=\begin{bmatrix}
\end{bmatrix}, \mathbf{b}=\begin{bmatrix}
a \\
Find all the values for $a$ so that the vector $\mathbf{b}$ is a linear combination of vectors $\mathbf{a}_1$ and $\mathbf{a}_2$.
Find the inverse matrix of
0 & 0 & 2 & 0 \\
0 &1 & 0 & 0 \\
1 & 0 & 0 & 1
\end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason.
Consider the system of linear equations
3x_1+2x_2&=1\\
5x_1+3x_2&=2.
(a) Find the coefficient matrix $A$ of the system.
(b) Find the inverse matrix of the coefficient matrix $A$.
(c) Using the inverse matrix of $A$, find the solution of the system.
(Linear Algebra Midterm Exam 1, the Ohio State University)
This post is Part 1 and contains the first three problems.
Problem 1.	Determine all possibilities for the number of solutions of each of the systems of linear equations described below.
(a) A consistent system of $5$ equations in $3$ unknowns and the rank of the system is $1$.
(b) A homogeneous system of $5$ equations in $4$ unknowns and it has a solution $x_1=1$, $x_2=2$, $x_3=3$, $x_4=4$.
Problem 2. Consider the homogeneous system of linear equations whose coefficient matrix is given by the following matrix $A$. Find the vector form for the general solution of the system.
1 & 0 & -1 & -2 \\
2 &1 & -2 & -7 \\
0 & 1 & 0 & -3
Problem 3. Let $A$ be the following invertible matrix.
-1 & 2 & 3 & 4 & 5\\
6 & -7 & 8& 9& 10\\
11 & 12 & -13 & 14 & 15\\
16 & 17 & 18& -19 & 20\\
21 & 22 & 23 & 24 & -25
\] Let $I$ be the $5\times 5$ identity matrix and let $B$ be a $5\times 5$ matrix.
Suppose that $ABA^{-1}=I$.
Then determine the matrix $B$.
An Example of Matrices $A$, $B$ such that $\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B)$
For an $m\times n$ matrix $A$, we denote by $\mathrm{rref}(A)$ the matrix in reduced row echelon form that is row equivalent to $A$.
For example, consider the matrix $A=\begin{bmatrix}
1 & 1 & 1 \\
0 &2 &2
\end{bmatrix}$
Then we have
\xrightarrow{\frac{1}{2}R_2}
0 &1 & 1
\xrightarrow{R_1-R_2}
\end{bmatrix}\] and the last matrix is in reduced row echelon form.
Hence $\mathrm{rref}(A)=\begin{bmatrix}
Find an example of matrices $A$ and $B$ such that
\[\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B).\]
Every Group of Order 24 Has a Normal Subgroup of Order 4 or 8
Prove that every group of order $24$ has a normal subgroup of order $4$ or $8$.
The Powers of the Matrix with Cosine and Sine Functions
Prove the following identity for any positive integer $n$.
\[\begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta& \cos \theta
\end{bmatrix}^n=\begin{bmatrix}
\cos n\theta & -\sin n\theta\\
\sin n\theta& \cos n\theta
Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.
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Lower and Upper Bounds of the Probability of the Intersection of Two Events
Square Root of an Upper Triangular Matrix. How Many Square Roots Exist?
True or False Problems on Midterm Exam 1 at OSU Spring 2018
Column Vectors of an Upper Triangular Matrix with Nonzero Diagonal Entries are Linearly Independent
Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials
Solve the System of Linear Equations Using the Inverse Matrix of the Coefficient Matrix
How to Diagonalize a Matrix. Step by Step Explanation.
Prove Vector Space Properties Using Vector Space Axioms
Express a Vector as a Linear Combination of Other Vectors
How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix
Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis
Matrix of Linear Transformation with respect to a Basis Consisting of Eigenvectors
12 Examples of Subsets that Are Not Subspaces of Vector Spaces
The Intersection of Two Subspaces is also a Subspace
Find a Basis and the Dimension of the Subspace of the 4-Dimensional Vector Space
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