Orthogonal Complements and Minimization Problems

Suppose $v_{1}, \ldots , v_{m} \in V$. Prove that

Suppose $U$ is a subspace of $V$ with basis $u_{1}, \ldots , u_{m}$ and

is a basis of $V$. Prove that if the Gram-Schmidt procedure is applied to the basis of $V$ above, producing a list $e_{1}, \ldots , e_{m}, f_{1}, \ldots , f_{n}$, then $e_{1}, \ldots , e_{m}$ is an orthonormal basis of $U$ and $f_{1}, \ldots , f_{n}$ is an orthonormal basis of $U^{\perp }$.

Suppose $U$ is the subspace of $\mathbb {R}^{4}$ defined by

Find an orthonormal basis of $U$ and an orthonormal basis of $U^{\perp }$.

Suppose $e_{1}, \ldots , e_{n}$ is a list of vectors in $V$ with $\left\| e_{k}\right\| = 1$ for each $k = 1, \ldots , n$ and

for all $v \in V$. Prove that $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $V$. This exercise provides a converse to $6.30(b)$.

Suppose that $V$ is finite-dimensional and $U$ is a subspace of $V$. Show that $P_{U^{\perp }} = I-P_{U}$, where $I$ is the identity operator on $V$.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V, W)$. Show that

Suppose that $X$ and $Y$ are finite-dimensional subspaces of $V$. Prove that $P_{X} P_{Y} = 0$ if and only if $\left\langle x, y \right\rangle = 0$ for all $x \in X$ and all $y \in Y$.

Suppose $U$ is a finite-dimensional subspace of $V$ and $v \in V$. Define a linear functional $\varphi : U \rightarrow \mathbf{F}$ by

for all $u \in U$. By the Riesz representation theorem (6.42) as applied to the inner product space $U$, there exists a unique vector $w \in U$ such that

for all $u \in U$. Show that $w = P_{U} v$.

Suppose $V$ is finite-dimensional. Suppose $P \in \mathcal{L}(V)$ is such that $P^{2} = P$ and every vector in $\operatorname {null}P$ is orthogonal to every vector in range $P$. Prove that there exists a subspace $U$ of $V$ such that $P = P_{U}$.

Suppose $V$ is finite-dimensional and $P \in \mathcal{L}(V)$ is such that $P^{2} = P$ and

for every $v \in V$. Prove that there exists a subspace $U$ of $V$ such that $P = P_{U}$.

Suppose $T \in \mathcal{L}(V)$ and $U$ is a finite-dimensional subspace of $V$. Prove that

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, and $U$ is a subspace of $V$. Prove that

Suppose $\mathbf{F} = \mathbb {R}$ and $V$ is finite-dimensional. For each $v \in V$, let $\varphi_{v}$ denote the linear functional on $V$ defined by

for all $u \in V$.

Suppose that $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $V$. Explain why the dual basis (see 3.112) of $e_{1}, \ldots , e_{n}$ is $e_{1}, \ldots , e_{n}$ under the identification of $V^{\prime }$ with $V$ provided by the Riesz representation theorem (6.58).

Definition 3.112 (dual basis): if $v_{1}, \ldots , v_{n}$ is a basis of $V$, then the dual basis of $v_{1}, \ldots , v_{n}$ is the list $\varphi_{1}, \ldots , \varphi_{n}$ of elements of $V^{\prime }$, where each $\varphi_{j}$ is the linear functional on $V$ such that $\varphi_{j}(v_{k}) = 1$ if $k = j$ and $\varphi_{j}(v_{k}) = 0$ if $k \neq j$.

Theorem 6.58 (Riesz representation theorem, revisited): suppose $V$ is finite-dimensional. For each $v \in V$, define $\varphi_{v} \in V^{\prime }$ by $\varphi_{v}(u) = \left\langle u, v \right\rangle$ for each $u \in V$. Then $v \mapsto \varphi_{v}$ is a one-to-one function from $V$ onto $V^{\prime }$.

In $\mathbb {R}^{4}$, let

Find $u \in U$ such that $\left\| u-(1,2,3,4)\right\|$ is as small as possible.

Suppose $C[-1,1]$ is the vector space of continuous real-valued functions on the interval $[-1,1]$ with inner product given by

for all $f, g \in C[-1,1]$. Let $U$ be the subspace of $C[-1,1]$ defined by

Find $p \in \mathcal{P}_{3}(\mathbb {R})$ such that $p(0) = 0, p^{\prime }(0) = 0$, and $\int_{0}^{1}|2+3 x-p(x)|^{2} d x$ is as small as possible.

Find $p \in \mathcal{P}_{5}(\mathbb {R})$ that makes $\int_{-\pi }^{\pi }|\sin x-p(x)|^{2} d x$ as small as possible. The polynomial 6.65 is an excellent approximation to the answer to this exercise, but here you are asked to find the exact solution, which involves powers of $\pi$. A computer that can perform symbolic integration should help.

Suppose $V$ is finite-dimensional and $P \in \mathcal{L}(V)$ is an orthogonal projection of $V$ onto some subspace of $V$. Prove that $P^{\dagger } = P$.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V, W)$. Show that

Suppose $T \in \mathcal{L}\left(\mathbf{F}^{3}, \mathbf{F}^{2}\right)$ is defined by

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V, W)$. Prove that

Both formulas above clearly hold if $T$ is invertible because in that case we can replace $T^{\dagger }$ with $T^{-1}$.

Suppose $V$ and $W$ are finite-dimensional and $T \in \mathcal{L}(V, W)$. Prove that

The equation above is analogous to the equation $\left(T^{-1}\right)^{-1} = T$ that holds if $T$ is invertible.