## 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*)