I have read everywhere that the resolution of the linear system $$Ax+b=0$$ where $A\in S_n(\mathbb R)$ and $b\in \mathbb R^n$ is equivalent to the resolution of the following optimization problem: $$f(x)=\frac 12\langle Ax,x\rangle+\langle b,x\rangle.$$ Which means that if $x\in \mathbb R^...

I need help making progress on a linear algebra question. Consider the subset $W = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \end{bmatrix}$ such that $x_1 + x_2 + x_3 = 2$ Does $x = \begin{bmatrix} 1 \\ 0\\ 1\\ \end{bmatrix} \in W?$ Does $u = \begin{bmatrix} 1\\ 1\\ 1\\ \end{bmat...

Show that if $V$ is a nonzero vector space over $\mathbb{R}$, and $V_{1}, V_{2},\ldots V_{k}$ are proper subspaces of $V$ then there exists $v\in V$ such that $v\not\in V_{i}$ for any $1\leq i\leq k$ I can prove the case for $k=2$ but I cannot produce an induction argument.

Update. The subspaces tag has been removed thanks to a friendly neighbourhood Community Manager. There's another possibility that hasn't been mentioned in the OP, and the one I would prefer: kill it! I don't see subspaces as a tag that will ever be consistently used, and not something that ...