(1) let $v = \mathcal{c}(-1,1)$ is a vector space over $\mathbb{r}$, let $f, h \in v$ defined by $f : x \mapsto x^2 , h: x \mapsto x|x|$ , is $\{f, h\}$ linearly independent ? (2) for which values of $t \in \mathbb{r}$ the subset $$ \left\{ \begin{pmatrix}\cos t+i \sin t \\ 1 \\ \end{pmatrix},\begin{pmatrix}1 \\ \cos t- i \sin t\\ \end{pmatrix}\right\}$$of $\mathbb{c}^2$ is linearly dependent ? (3) consider e1 and e2 are projections on m1 and m2 along n1 and n2 respectively. verify the statement: if e1 and e2 commute, then e1 + e2 − e1e2 is a projection (4) let v be a vector space over f and let α; β be projections on v satisfying the condition that im(α) and im(β) are disjoint. is it necessarily true that αβ = βα? (5) let v = c(−1; 1) and let $a > -1/2$ be a real number. is the function $\alpha : v \times v \rightarrow \mathbb{r}$ defined by $\alpha(f, g) =\int_{-1}^1 (1-t^2)^{a-1/2}f(t)g(t) dt $ an inner product on v?