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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 $7 \times 5$.
