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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