Find the span of $U=\{2,\cos x,\sin x:x\in\mathbb{R}\}$ ($U$ is the subset of a space of real functions) and $V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\}$
Attempt:
Objects in $U$ :$2,\cos x,\sin x$ are linearly independent, so they span $\mathbb{R^3}$.
Let ,$n=3\Rightarrow [V]= \begin{bmatrix} a & b & b \\ b & a & b \\ b & b & a \\ \end{bmatrix}$
$rref[V]=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}\Rightarrow $ vectors in $V$ span $\mathbb{R^3}$, if $a,b\neq 0$.
But because $V\subset\mathbb{R^n}\Rightarrow $ vectors span $\mathbb{R^{n-1}}$.
Is this correct?