Answer
a. True.
b. False.
c. True.
d. False.
Work Step by Step
a. Assuming that the two vectors are not both the zero vector, we know from the box on page 59 that two vectors are linearly independent if and only if neither vector is a multiple of the other. Hence, two vectors are linearly dependent if and only if at least one is a constant multiple of the other. Since the constant multiples of a vector span a line through the origin, it must be that the two vectors line on the same line through the origin.
b. Theorem 8 states that, if a set contains more vectors than entries in each vector, then the set is linearly dependent. However, the inverse of this theorem is not true, and so it is certainly not the case that this stricter variation of the inverse--that a set with fewer (as opposed to fewer or equal) vectors than entries is linearly independent--need be true.
c. By definition, $\vec{z}$ is in $Span\{\vec{x},\vec{y}\}$ if and only if $\vec{z}$ is a linear combination of $\vec{x}$ and $\vec{y}$. Further, Theorem 7 tells us that a set of vectors is linearly dependent if and only if one of the vectors in a set is a linear combination of previous vectors in the set (unless the first vector is the zero vector, which by Theorem 9 contradicts our hypothesis that $\vec{x}$ and $\vec{y}$ are linearly independent). Hence, the set $\{\vec{x},\vec{y},\vec{z}\}$ must be linearly dependent.
d. Theorem 8 states that, if a set contains more vectors than entries in each vector, then the set is linearly dependent. However, the converse--that a linearly dependent set of vectors contains more vectors than entries in each vector--is not necessarily true.
