Define standard vector addition, and scalar multiplication for \(\mathbb{R}^3\). Write down the zero vector and \(-\mathbf{x}\) where \(\mathbf{x} = \left(\begin{array}{c}x\\y\\z\end{array}\right)\).
Prove that \(\alpha\cdot \mathbf{0} = \mathbf{0}\) and \(0\cdot \mathbf{v} = \mathbf{0}\).
Let \(V = F[x]\) and let \(W = \{ f\in V\ \mid\ f(-x) = f(x)\}\) be the subset of even polynomials with coefficients in \(F\). Prove that \(W\) is a subspace of \(V\). If \(F = \mathbb{R}\) prove that \(W = \mathrm{span}(1, x^2, x^4,\ldots, x^{2n},\ldots)\).
There is an error in the following example from the video: \(W = \{ A\in \mathbb{C}^{n\times n} \ \mid \ A^* = A\}\). What is the error?
Let \(X = \{0,1,2,3,4\}\) and \(V_1 = \mathbb{R}^X\) and \(V_2 = \mathbb{R}^5\). Can you find a correspondence between \(V_1\) and \(V_2\)?