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| A matrix represents a linear transformation of vectors. Is the reverse true? Namely, can any linear transformation (of a finite dimensional vectors) be represented by a matrix? | A matrix represents a linear transformation of vectors. Is the reverse true? Namely, can any linear transformation (of finite dimensional vectors) be represented by a matrix? |
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| Ans: A |
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| Suppose that a linear transformation of a 2D vector space maps $\bigl(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix} \bigr)$ to $\vec{a}$, and $\bigl(\begin{smallmatrix} 0 \\ 1 \end{smallmatrix} \bigr)$ to $\vec{b}$. The matrix that represents this linear transformation is | Suppose that a linear transformation of a 2D vector space maps $\bigl(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix} \bigr)$ to $\vec{a}$, and $\bigl(\begin{smallmatrix} 0 \\ 1 \end{smallmatrix} \bigr)$ to $\vec{b}$. (Recall that we consider $\vec{a}$ and $\vec{b}$ as column vectors.) The matrix that represents this linear transformation is |
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| A. $\bigl(\vec{a}\quad \vec{b}\bigr)$ A. $\bigl(\begin{smallmatrix} \vec{a}^t \\ \vec{b}^t \end{smallmatrix} \bigr)$ |
A. $\left(\vec{a}\quad \vec{b}\right)$ A. $\left(\begin{smallmatrix} \; \vec{a}^t \; \\ \; \vec{b}^t \; \end{smallmatrix} \right)$ |
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Ans: A |
Linear transformation and matrix
A matrix represents a linear transformation of vectors. Is the reverse true? Namely, can any linear transformation (of finite dimensional vectors) be represented by a matrix?
- Yes
- No
Ans: A
Matrix and linear transformation
Suppose that a linear transformation of a 2D vector space maps $\bigl(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix} \bigr)$ to $\vec{a}$, and $\bigl(\begin{smallmatrix} 0 \\ 1 \end{smallmatrix} \bigr)$ to $\vec{b}$. (Recall that we consider $\vec{a}$ and $\vec{b}$ as column vectors.) The matrix that represents this linear transformation is
$\left(\vec{a}\quad \vec{b}\right)$
$\left(\begin{smallmatrix} \; \vec{a}^t \; \\ \; \vec{b}^t \; \end{smallmatrix} \right)$
- None of the above.
Ans: A