Let
$U_1 = (1, 1, 0, 0)$
$U_2 = (1, 0, 1, 0)$
$U_3 = (0, 0, 1, 1)$
Without doing any computation, explain why span$\{U_1, U_2, U_3\}$ $\neq$ $\mathbb{R}^4$.
I could reduce it to reduced row echelon form via Gaussian Elimination but that will not satisfy the condition of 'without doing any computation'. Is there any way I can do this problem without computation?
On
An argument based on dimension is the simplest way to go. If you don’t like that approach, here’s another, although it may not qualify as “without computation.”
Consider the vector $(0,0,0,1)$. What linear combination of the given vectors can produce it? There must be a non-zero multiple of $U_3$ since that’s the only one with a non-zero fourth coordinate. This means that you also have to have a non-zero multiple of $U_2$ to cancel the third coordinate that $U_3$ brings with it, which in turn requires a non-zero multiple of $U_1$ to cancel the first coordinate that came with $U_2$, but now you’re stuck with no way to cancel the non-zero second coordinate of $U_1$ since it’s zero in the other two vectors. So, you have an element of $\mathbb R^4$ that’s not in the span of the three given vectors.
The space $\mathbb R^4$ has dimension four.