Exercise 3.4.9

Answers

(a): The four vectors in $R^{3}$ are the columns of a 3 by 4 matrix $A$ . There is a nonzero solution to $Ax = 0$ because there is at least one free variable;
(b): Two vectors are dependent if $[\begin{matrix} v_{1} & v_{2} \end{matrix}]$ has rank 0 or 1 . (OK to say "they are on the same line" or "one is a multiple of the other" but not " $v_{2}$ is a multiple of $v_{1} ” - since v_{1}$ might be $0 .)$
(c): A nontrivial combination of $v_{1}$ and $0$ gives $0 : 0 v_{1} + 3 (0, 0, 0) = 0$ .

mathchakra

2022-01-23 15:44