Bilinear, Quadratic Forms, and Inner Product Spaces

Let $f: \mathbb {R}^{n} \to \mathbb {R}$ be a function such that

Let $A, B, \ldots , F$ be real coefficients. Show that the quadratic form

is positive definite if and only if

Find all real numbers $t$ for which the quadratic form $Q_{t}$ on $\mathbb {R}^{3}$, defined by

is positive definite.

Let $\mathbb {R}^{3}$ be $3$-space with the usual inner product, and $(a,b,c) \in \mathbb {R}^{3}$ a vector of length $1$. Let $W$ be the plane defined by $ax+by+cz = 0$. Find, in the standard basis, the matrix representing the orthogonal projection of $\mathbb {R}^{3}$ onto $W$.

Let $w$ be a positive continuous function on $[0,1]$, $n$ a positive integer, and $P_{n}$ the vector space of real polynomials whose degrees are at most $n$, equipped with the inner product

For continuous real valued functions $f, g$ on the interval $[-1,1]$ define the inner product $\left\langle f, g \right\rangle = \int_{-1}^{1} f(x)g(x) \, dx$. Find that polynomial of the form $p(x) = a+bx^{2}-x^{4}$ which is orthogonal on $[-1,1]$ to all lower order polynomials.

Let $E$ be a finite-dimensional vector space over a field $\mathbf{F}$. Suppose $B: E \times E \to \mathbf{F}$ is a bilinear map (not necessarily symmetric). Define subspaces

Prove that $\dim E_{1} = \dim E_{2}$.

Let $A$ be a real $n \times n$ matrix such that $\left\langle Ax, x \right\rangle \geq 0$ for every real $n$-vector $x$. Show that $Au = 0$ if and only if $A^{t}u = 0$.

An $n \times n$ real matrix $T$ is positive definite if $T$ is symmetric and $\left\langle Tx, x \right\rangle > 0$ for all nonzero vectors $x \in \mathbb {R}^{n}$, where $\left\langle u, v \right\rangle$ is the standard inner product. Suppose that $A$ and $B$ are two positive definite real matrices.

Let $V$ be a real vector space of dimension $n$, and let $S: V \times V \to \mathbb {R}$ be a nondegenerate bilinear form. Suppose that $W$ is a linear subspace of $V$ such that the restriction of $S$ to $W \times W$ is identically $0$. Show that $\dim W \leq n/2$.

Let $A$ be the symmetric matrix

Denote by $v$ the column vector

in $\mathbb {R}^{3}$, and by $x^{t}$ its transpose $(x,y,z)$. Let $\left\| v\right\|$ denote the length of the vector $v$. As $v$ ranges over the set of vectors for which $v^{t}Av = 1$, show that $\left\| v\right\|$ is bounded, and determine its least upper bound.

Define the index of a real symmetric matrix $A$ to be the number of strictly positive eigenvalues of $A$ minus the number of strictly negative eigenvalues. Suppose $A$ and $B$ are real symmetric $n \times n$ matrices such that $x^{t}Ax \leq x^{t}Bx$ for all $n \times 1$ matrices $x$. Prove the index of $A$ is less than or equal to the index of $B$.

For $x, y \in \mathbb {C}^{n}$, let $\left\langle x, y \right\rangle$ be the Hermitian inner product $\sum_{j} x_{j}\overline{y_{j}}$. Let $T$ be a linear operator on $\mathbb {C}^{n}$ such that $\left\langle Tx, Ty \right\rangle = 0$ if $\left\langle x, y \right\rangle = 0$. Prove that $T = kS$ for some scalar $k$ and some operator $S$ which is unitary: $\left\langle Sx, Sy \right\rangle = \left\langle x, y \right\rangle$ for all $x$ and $y$.

Let $E$ denote a finite-dimensional complex vector space with a Hermitian inner product $\left\langle x, y \right\rangle$.

Let $a$ and $b$ be real numbers. Prove that there are two orthogonal unit vectors $u$ and $v$ in $\mathbb {R}^{3}$ such that $u = (u_{1}, u_{2}, a)$ and $v = (v_{1}, v_{2}, b)$ if and only if $a^{2}+b^{2} \leq 1$.