I have the following vectors $$ u_1 = (1, t, 2t, 0)^T, u_2 = (t, t, t, t)^T, v_1 = (t - 2, -t, -3t, t)^T, v_2 = (2, t, 2t, 0)^T $$
and
$$ U = span\{u_1, u_2\} , V = span\{v_1, v_2\} $$
And I am to find a $t$ so that $U + V = \mathbb R^4$
To be honest I'm not entirely sure where to start. I have tried to understand the definition of $U + V$ but what I'm unsure of is if I have to add every vector in $U$ with every vector in $V$ so that
\begin{align*}U + V &= span\{u_1 + v_1, u_1 + v_2, u_2 + v_1, u_2 + v_2\} \\ &=span\{(t-1, 0, -t, t)^T, (3, 2t, 4t, 0)^T, (2t-2, 0, -2t, 2t)^T, (t+2, 2t, 3t, t)^T\} \end{align*} and then put that into a matrix and more or less solve for t, or if I have perhaps completely misunderstood what $U + V$ actually means.
We define the direct sum of two vector spaces, $U,V$, denoted $U+V$ or $U\oplus V$ to be be the set $W$ such that $W=\{w=u+v:u\in U,v\in V\}$. In words, this set is what you get when you take each element of $U$ and add it to each element of $V$.
It is a fact about direct sums that if $G_U$ generates $U$ and $G_V$ generates $V$ then $G_{U+V}=G_U\cup G_V$ generates $U+V$. Therefore you’re interested in finding the values of $t$ such that $span\{u_1,u_2,v_1,v_2\}=\mathbb R^4$