How do I check if a vector space spans a particular vector?

Question

I have been given $5$ vectors $(u_1, u_2, u_3, u_4 \text{ and } u_5)$:

$u_1= \langle1,-1,2,1\rangle, u_2 = \langle1,2,1,-1\rangle, u_3 = \langle-1,-8,1,5\rangle, u_4 = \langle1,1,1,1\rangle, u_5 = \langle-7,2,4,1\rangle$.

I have a vector space $V$ that is the span of the $5$ vectors. So

$V = Span\{u_1, u_2, u_3, u_4\}$. The vector space $V$ has $3$ dimensions.

I have to check if $V^\perp = Span\{u_5\}$. How do I do that?

First I wrote $u_5$ as $v = \langle-7t, 2t, 4t, t\rangle$. Next I found that this vector is orthogonal to all the other vectors $(u_1, u_2, u_3, u_4 \text{ and } u_5)$. I'm not sure how to continue. I'm not even sure if I'm doing it right.

Answer 1

I am assuming you meant that $V = \mbox{span}\{u_1,u_2,u_3,u_4\}$.

If $u_5 \perp u_i$ for all $i \in \{1,2,3,4\}$, then $u_5 \perp \mbox{span}\{u_1,u_2,u_3,u_4\} = V$. The span of a set of vectors is just the set of all finite linear combinations of them, and so if the inner product (dot product) of $u_5$ with each of these vectors is 0, then its inner product with any vector in their span is 0 as well.

Answer 2

You check that u5 is orthogonal to u1 u2 u3 and u4 as a proof. Or you can equivalently prove that u5 is orthogonal to the 3 dimensions you found of V. Make sure to use the given inner product.