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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Quiz 11. Find Eigenvalues and Eigenvectors/ Properties of Determinants

Problem 363

(a) Find all the eigenvalues and eigenvectors of the matrix
\[A=\begin{bmatrix}
3 & -2\\
6& -4
\end{bmatrix}.\]

(b) Let
\[A=\begin{bmatrix}
1 & 0 & 3 \\
4 &5 &6 \\
7 & 0 & 9
\end{bmatrix} \text{ and } B=\begin{bmatrix}
2 & 0 & 0 \\
0 & 3 &0 \\
0 & 0 & 4
\end{bmatrix}.\] Then find the value of
\[\det(A^2B^{-1}A^{-2}B^2).\] (For part (b) without computation, you may assume that $A$ and $B$ are invertible matrices.)

 
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Quiz 10. Find Orthogonal Basis / Find Value of Linear Transformation

Problem 356

(a) Let $S=\{\mathbf{v}_1, \mathbf{v}_2\}$ be the set of the following vectors in $\R^4$.
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
0 \\
1 \\
0
\end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix}
0 \\
1 \\
1 \\
0
\end{bmatrix}.\] Find an orthogonal basis of the subspace $\Span(S)$ of $\R^4$.

 
(b) Let $T:\R^2 \to \R^3$ be a linear transformation such that
\[T(\mathbf{e}_1)=\mathbf{u}_1 \text{ and } T(\mathbf{e}_2)=\mathbf{u}_2,\] where $\{\mathbf{e}_1, \mathbf{e}_2\}$ is the standard unit vectors of $\R^2$ and
\[\mathbf{u}_1=\begin{bmatrix}
5 \\
1 \\
2
\end{bmatrix} \text{ and } \mathbf{u}_2=\begin{bmatrix}
8 \\
2 \\
6
\end{bmatrix}.\] Then find
\[T\left(\, \begin{bmatrix}
3 \\
-2
\end{bmatrix} \,\right).\]

 
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Find a General Formula of a Linear Transformation From $\R^2$ to $\R^3$

Problem 353

Suppose that $T: \R^2 \to \R^3$ is a linear transformation satisfying
\[T\left(\, \begin{bmatrix}
1 \\
2
\end{bmatrix}\,\right)=\begin{bmatrix}
3 \\
4 \\
5
\end{bmatrix} \text{ and } T\left(\, \begin{bmatrix}
0 \\
1
\end{bmatrix} \,\right)=\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix}.\] Find a general formula for
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix} \,\right).\]

(The Ohio State University, Linear Algebra Math 2568 Exam Problem)

 
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Hyperplane in $n$-Dimensional Space Through Origin is a Subspace

Problem 352

A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors
\[\begin{bmatrix}
x_1 \\
x_2 \\
\vdots \\
x_n
\end{bmatrix}\in \R^n\] satisfying the linear equation of the form
\[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\] where $a_1, a_2, \dots, a_n$ (at least one of $a_1, a_2, \dots, a_n$ is nonzero) and $b$ are real numbers.
Here at least one of $a_1, a_2, \dots, a_n$ is nonzero.

Consider the hyperplane $P$ in $\R^n$ described by the linear equation
\[a_1x_1+a_2x_2+\cdots+a_nx_n=0,\] where $a_1, a_2, \dots, a_n$ are some fixed real numbers and not all of these are zero.
(The constant term $b$ is zero.)

Then prove that the hyperplane $P$ is a subspace of $R^{n}$ of dimension $n-1$.

 
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Coordinate Vectors and Dimension of Subspaces (Span)

Problem 350

Let $V$ be a vector space over $\R$ and let $B$ be a basis of $V$.
Let $S=\{v_1, v_2, v_3\}$ be a set of vectors in $V$. If the coordinate vectors of these vectors with respect to the basis $B$ is given as follows, then find the dimension of $V$ and the dimension of the span of $S$.
\[[v_1]_B=\begin{bmatrix}
1 \\
0 \\
0 \\
0
\end{bmatrix}, [v_2]_B=\begin{bmatrix}
0 \\
1 \\
0 \\
0
\end{bmatrix}, [v_3]_B=\begin{bmatrix}
1 \\
1 \\
0 \\
0
\end{bmatrix}.\]

 
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Quiz 9. Find a Basis of the Subspace Spanned by Four Matrices

Problem 349

Let $V$ be the vector space of all $2\times 2$ real matrices.
Let $S=\{A_1, A_2, A_3, A_4\}$, where
\[A_1=\begin{bmatrix}
1 & 2\\
-1& 3
\end{bmatrix}, A_2=\begin{bmatrix}
0 & -1\\
1& 4
\end{bmatrix}, A_3=\begin{bmatrix}
-1 & 0\\
1& -10
\end{bmatrix}, A_4=\begin{bmatrix}
3 & 7\\
-2& 6
\end{bmatrix}.\] Then find a basis for the span $\Span(S)$.

 
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Condition that a Matrix is Similar to the Companion Matrix of its Characteristic Polynomial

Problem 348

Let $A$ be an $n\times n$ complex matrix.
Let $p(x)=\det(xI-A)$ be the characteristic polynomial of $A$ and write it as
\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\] where $a_i$ are real numbers.

Let $C$ be the companion matrix of the polynomial $p(x)$ given by
\[C=\begin{bmatrix}
0 & 0 & \dots & 0 &-a_0 \\
1 & 0 & \dots & 0 & -a_1 \\
0 & 1 & \dots & 0 & -a_2 \\
\vdots & & \ddots & & \vdots \\
0 & 0 & \dots & 1 & -a_{n-1}
\end{bmatrix}=
[\mathbf{e}_2, \mathbf{e}_3, \dots, \mathbf{e}_n, -\mathbf{a}],\] where $\mathbf{e}_i$ is the unit vector in $\C^n$ whose $i$-th entry is $1$ and zero elsewhere, and the vector $\mathbf{a}$ is defined by
\[\mathbf{a}=\begin{bmatrix}
a_0 \\
a_1 \\
\vdots \\
a_{n-1}
\end{bmatrix}.\]

Then prove that the following two statements are equivalent.

  1. There exists a vector $\mathbf{v}\in \C^n$ such that
    \[\mathbf{v}, A\mathbf{v}, A^2\mathbf{v}, \dots, A^{n-1}\mathbf{v}\] form a basis of $\C^n$.
  2. There exists an invertible matrix $S$ such that $S^{-1}AS=C$.
    (Namely, $A$ is similar to the companion matrix of its characteristic polynomial.)

 
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Linearly Dependent if and only if a Vector Can be Written as a Linear Combination of Remaining Vectors

Problem 347

Let $V$ be a vector space over a scalar field $K$.
Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ be the set of vectors in $V$, where $n \geq 2$.

Then prove that the set $S$ is linearly dependent if and only if at least one of the vectors in $S$ can be written as a linear combination of remaining vectors in $S$.

 
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