# 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.

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

- Problem 364
- Solution.
- (1) True or False? Let $A$ be a $5\times 3$ matrix. Then the range of $A$ is a subspace in $\R^3$.
- (2) True or False? The function $f(x)=x^2+1$ is not in the vector space $C[-1,1]$ because $f(0)=1\neq 0$.
- (3) True or False? Since we have $\sin(x+y)=\sin(x)+\sin(y)$, the function $\sin(x)$ is a linear transformation.
- (4) True or False? The given set is an orthonormal set.

- Linear Algebra Midterm Exam 2 Problems and Solutions

## Solution.

### (1) True or False? Let $A$ be a $5\times 3$ matrix. Then the range of $A$ is a subspace in $\R^3$.

The answer is “False”. The definition of the range of the $5 \times 3$ matrix $A$ is

\[ \calR(A)=\{\mathbf{y}\in \R^5 \mid A\mathbf{x}=\mathbf{y} \text{ for some $\mathbf{x} \in \R^3$}\}.\]
Note that to make sense the matrix product $A\mathbf{x}$, the size of the vector $\mathbf{x}$ must be $3$-dimensional because $A$ is $5\times 3$. Hence $\mathbf{y}=A\mathbf{x}$ is a $5$-dimensional vector, and thus the range $\calR(A)$ is a subspace of $\R^5$.

### (2) True or False? The function $f(x)=x^2+1$ is not in the vector space $C[-1,1]$ because $f(0)=1\neq 0$.

The answer is “False”. The vector space $C[-1, 1]$ consists of all continuos functions defined on the interval $[-1, 1]$. Since $f(x)=x^2+1$ is a continuos function defined on $[-1, 1]$, it is in the vector space $C[-1, 1]$. The condition $f(0)=1\neq 0$ is irrelevant.

### (3) True or False? Since we have $\sin(x+y)=\sin(x)+\sin(y)$, the function $\sin(x)$ is a linear transformation.

The answer is “False”. First of all $\sin(x+y)\neq \sin(x)+\sin(y)$. For example, let $x=y=\pi/2$. Then

\[\sin\left(\,\frac{\pi}{2}+\frac{\pi}{2}\,\right)=\sin\left(\,\pi \,\right)=0\]
and

\[\sin\left(\, \frac{\pi}{2} \,\right)+\sin\left(\, \frac{\pi}{2} \,\right)=1+1=2.\] Hence $\sin(x)$ is not a linear transformation.

### (4) True or False? The given set is an orthonormal set.

The answer is “False”. The dot product of these vectors is

\[\begin{bmatrix}

1 \\

0 \\

0

\end{bmatrix}\cdot \begin{bmatrix}

0 \\

1 \\

1

\end{bmatrix}=1\cdot 0+ 0\cdot 1 +0\cdot 1=0.\]
Thus, the vectors are orthogonal. However the length of the second vector is

\[\sqrt{0^2+1^2+1^2}=\sqrt{2},\]
hence it is not the unit vector.

So the set is orthogonal, but not orthonormal set.

## Linear Algebra Midterm Exam 2 Problems and Solutions

- True of False Problems and Solutions (current problem): 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 4-dimensional vector space

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