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Linear Algebra

Linear Algebra Midterm 1 at the Ohio State University (3/3)

Problem 572

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 $A=\begin{bmatrix}
-3 & -4\\
8& 9
\end{bmatrix}$ and $\mathbf{v}=\begin{bmatrix}
-1 \\
2
\end{bmatrix}$.

(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
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix}\] are linearly independent.

 
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Ring theory

Is the Given Subset of The Ring of Integer Matrices an Ideal?

Problem 524

Let $R$ be the ring of all $2\times 2$ matrices with integer coefficients:
\[R=\left\{\, \begin{bmatrix}
a & b\\
c& d
\end{bmatrix} \quad \middle| \quad a, b, c, d\in \Z \,\right\}.\]

Let $S$ be the subset of $R$ given by
\[S=\left\{\, \begin{bmatrix}
s & 0\\
0& s
\end{bmatrix} \quad \middle | \quad s\in \Z \,\right\}.\]

(a) True or False: $S$ is a subring of $R$.

(b) True or False: $S$ is an ideal of $R$.

 
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Linear Algebra

10 True of False Problems about Nonsingular / Invertible Matrices

Problem 500

10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.

The quiz is designed to test your understanding of the basic properties of these topics.

You can take the quiz as many times as you like.

The solutions will be given after completing all the 10 problems.
Click the View question button to see the solutions.

 
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Linear Algebra

True of False Problems on Determinants and Invertible Matrices

Problem 438

Determine whether each of the following statements is True or False.

(a) If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$.

(b) If the characteristic polynomial of an $n \times n$ matrix $A$ is
\[p(\lambda)=(\lambda-1)^n+2,\] then $A$ is invertible.

(c) If $A^2$ is an invertible $n\times n$ matrix, then $A^3$ is also invertible.

(d) If $A$ is a $3\times 3$ matrix such that $\det(A)=7$, then $\det(2A^{\trans}A^{-1})=2$.

(e) If $\mathbf{v}$ is an eigenvector of an $n \times n$ matrix $A$ with corresponding eigenvalue $\lambda_1$, and if $\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_2$, then $\mathbf{v}+\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_1+\lambda_2$.

(Stanford University, Linear Algebra Exam Problem)
 
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Linear Algebra

True or False Problems of Vector Spaces and Linear Transformations

Problem 364

These are True or False problems.
For each of the following statements, determine if it contains a wrong information or not.

  1. Let $A$ be a $5\times 3$ matrix. Then the range of $A$ is a subspace in $\R^3$.
  2. The function $f(x)=x^2+1$ is not in the vector space $C[-1,1]$ because $f(0)=1\neq 0$.
  3. Since we have $\sin(x+y)=\sin(x)+\sin(y)$, the function $\sin(x)$ is a linear transformation.
  4. The set
    \[\left\{\, \begin{bmatrix}
    1 \\
    0 \\
    0
    \end{bmatrix}, \begin{bmatrix}
    0 \\
    1 \\
    1
    \end{bmatrix} \,\right\}\] is an orthonormal set.

(Linear Algebra Exam Problem, The Ohio State University)

 
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Linear Algebra

True or False. The Intersection of Bases is a Basis of the Intersection of Subspaces

Problem 253

Determine whether the following is true or false. If it is true, then give a proof. If it is false, then give a counterexample.

Let $W_1$ and $W_2$ be subspaces of the vector space $\R^n$.
If $B_1$ and $B_2$ are bases for $W_1$ and $W_2$, respectively, then $B_1\cap B_2$ is a basis of the subspace $W_1\cap W_2$.

 
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Linear Algebra

True or False Quiz About a System of Linear Equations

Problem 78

Determine whether the following sentence is True or False.

(Purdue University Linear Algebra Exam)

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