Math Problems Vault

Problem VI.5.1

(a)(b)

Can a linear function on a linear space $\mathcal{L}$ satisfy, for all $x \in \mathcal{L}$ , the inequality

?

(a)

$\mathrm{f}(x)>0$

(b)

$\mathrm{f}(x) \geqslant 0$ ?

Problem VI.5.2

Let $\vec{a}$ be a fixed vector of the plane. Assign to every vector $\vec{x}$ the area of the oriented parallelogram built on $\vec{x}$ and $\vec{a}$ , or 0 if the vectors are collinear. Verify that this function is linear, and find the row of its coefficients in the basis $\vec{e}_{1}, \vec{e}_{2}$ , if $\vec{a}=\alpha \vec{e}_{1}+\beta \vec{e}_{2}$ .

?

Problem VI.5.3

(a)(b)

Let $k$ be a natural number. Assign to every polynomial of degree at most $n$ the value of its $k$ -th derivative at the point $a$ . Verify that this defines a linear function. Find its coordinate row in the bases

?

(a)

$1, t, t^{2}, \ldots , t^{n}$

(b)

$1,(t-a),(t-a)^{2}, \ldots ,(t-a)^{n}$ .

Problem VI.5.4

(a)(b)

Let $e_{1}, \ldots , e_{n} \in \mathcal{L}$ and $\mathrm{e}^{* 1}, \ldots , \mathrm{e}^{* n} \in \mathcal{L}^{*}$ be a pair of biorthogonal bases. Verify that for any $x \in \mathcal{L}$ and for any $\mathrm{f} \in \mathcal{L}^{*}$ the equalities hold:

?

(a)

$x=\mathrm{e}^{* 1}(x) e_{1}+\ldots +\mathrm{e}^{* n}(x) e_{n}$ and

(b)

$\mathrm{f}=\mathrm{f}\left(e_{1}\right) \mathrm{e}^{* 1}+\ldots +\mathrm{f}\left(e_{n}\right) \mathrm{e}^{* n}$ .