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Problems #35 on pg.61 and #8 on pg.68 are graded.

#35,pg.61: TRUE OR FALSE? If

$\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_5$ are in

$\mathbb{R}^{5 \times 1}$ and

$\vec{v}_3=\vec{0}$ , then

$\{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_5\}$ is linearly dependent.
Solution: This is true because of Theorem 9, pg.59, which states that any set of vectors containing

$\vec{0}$ is a linearly dependent set of vectors.

#8,pg.68: How many rows and columns must a matrix

$A$ have in order to define a mapping form

$\mathbb{R}^{5 \times 1}$ into

$\mathbb{R}^{7 \times 1}$ by the rule

$T(\vec{x})=A\vec{x}$ ?
Solution: Let us write the size of

$A$ as

$m \times n$ and the size of

$\vec{x}=\ell \times 1$ . We know that the multiplication

$A\vec{x}$ is well-defined if and only if

$n=\ell$ . We would like the "input" vectors

$\vec{x}$ to be of size

$5 \times 1$ and the "output" vectors to be of size

$7 \times 1$ . Therefore

$\vec{x}$ must be of size

$5 \times 1$ and

$A$ must be of size

$7 \times 5$ .