Math Problems Vault

Problem VII.1.1

(a)(b)(c)

Verify that in the space of polynomials of degree $\leqslant 2$ an inner product can be defined by the formula

(p, q)=\int _{-1}^{1} p(t) q(t) d t

?

(a)

Write the Gram matrix of the basis 1, $t, t^{2}$ .

(b)

Using the change-of-basis matrix, find the Gram matrix of the basis 1, $(t-1)$ , $(t-1)^{2}$ .

(c)

Find the angle between the polynomials $t^{2}+1$ and $t+1$ .

Problem VII.1.2

A subspace of a Euclidean space is given in an orthonormal basis by the equation $\xi^{1}+\xi^{2}+\xi^{3}+\xi^{4}=0$ . Find an orthonormal basis in this subspace.

?

Problem VII.1.3

(a)(b)

Let $\operatorname {dim} \mathcal{E}=4$ and let $\mathcal{E}^{\prime } \subseteq \mathcal{E}$ be given in an orthonormal basis by the system

\xi _{1}+\xi _{2}+\xi _{3}=0, \xi _{2}+\xi _{3}+\xi _{4}=0 .

Find:

?

(a)

a basis in $\mathcal{E}^{\prime \perp }$

(b)

the orthogonal projection onto $\mathcal{E}^{\prime }$ of the vector $a$ with coordinates $\mathbf{a}=\| 1,2,3,4\|^{T}$ .

Problem VII.1.4

(a)(b)

Suppose that all the entries of an orthogonal matrix of order $n$ are equal to one another in absolute value.

?

(a)

What is the absolute value of an entry of such a matrix?

(b)

Prove that such matrices exist if $n=2^{k}$ , where $k$ is a natural number.

Problem VII.1.5

(a)(b)

Find the $Q R$ decomposition of the matrix

?

(a)

$A=\left\| \begin{array}{ll}1 & 1 \\ 2 & 3\end{array}\right\|$

(b)

$B=\left\| \begin{array}{rrr}1 & 3 & 1 \\ 2 & 4 & -3 \\ 1 & 1 & -1\end{array}\right\|$ .

Problem VII.1.6

Prove that the $Q R$ decomposition of a matrix is unique.

?

Problem VII.1.7

In a four-dimensional Euclidean space a three-dimensional parallelepiped is built on the vectors having in an orthonormal basis the coordinate columns $\| 1,1,-1,0\|^{T}$ , $\| 1,1,1,-1\|^{T}$ and $\| 1,1,1,1\|^{T}$ . Find the volume of the parallelepiped.

?