Question:
a. show that the vectors u = {(1,1,0,0), (0,1,1,0), (0,0,1,1), (1,0,0,1)}={$v_1$, $v_2$, $v_3$, $v_4$} is a basis in $R^4$.
b. the function f(v) = $[.]_u$ given by $[v]_u$ = ($a_1$, $a_2$, $a_3$, $a_4$) with v = $a_1v_1$+ $a_2v_2$+ $a_3v_3$+$a_4v_4$ is linear. compute the matrix representation of f with respect to the standard basis, [f].
Attempt:
a. the set u is a basis of $R^4$ if the vectors are linearly independent. so I put the vectors in matrix form and check whether they are linearly independent.
$$ \left[
\begin{array}{cccc}
1&0&0&1\\
1&1&0&0\\
0&1&1&0\\
0&0&1&1
\end{array}
\right] $$
so i tried to put the matrix in RREF this is what I got.
$$ \left[
\begin{array}{cccc}
1&0&0&1\\
0&1&0&-1\\
0&0&1&1\\
0&0&0&0
\end{array}
\right] $$
we can see that the set is not linearly independent therefore it does not span $R^4$. IS THAT CORRECT?
b. how do i solve b.
thanks
Best Answer
Notice that $(1,0,0,1)-(0,0,1,1)+(0,1,1,0)=(1,1,0,0)$ so they are not linearly independent. Since $\mathbb R^4$ has dimension $4$, you need $4$ nonzero linearly independent vectors to form a basis, so you're correct.