Let $\mathbf{x}={\left(-1,-1,1,1\right)}^T$ and $\mathbf{y}={\left(1,1,5,-3\right)}^T$. Show that $\mathbf{x}\perp \mathbf{y}$. Calculate ${\Vert \mathbf{\text{x}}\Vert }_2,{\Vert \mathbf{y}\Vert }_2,{\Vert \mathbf{x}+\mathbf{y}\Vert }_2$ and verify that the Pythagorean law holds.

Let $\mathbf{x}={\left(1,1,1,1\right)}^T$ and $\mathbf{y}={\left(8,2,2,0\right)}^T$.

1. Determine the angle $\theta$ between x and y.

2. Find the vector projection p of x onto y.

3. Verify that $\mathbf{x}-\mathbf{p}$ is orthogonal to p.

4. Compute ${\Vert \mathbf{x}-\mathbf{p}\Vert }_2,{\Vert \mathbf{p}\Vert }_2,{\Vert \mathbf{x}\Vert }_2$ and verify that the Pythagorean law is satisfied.

Use equation (1) with weight vector $\mathbf{w}={\left(\frac{1}{4},\frac{1}{2},\frac{1}{4}\right)}^T$ to define an inner product for ${ℝ}^3$, and let $\mathbf{x}={\left(1,1,1\right)}^T$ and $\mathbf{y}={\left(-5,1,3\right)}^T$.

1. Show that x and y are orthogonal with respect to this inner product.

2. Compute the values of $\Vert \mathbf{x}\Vert$ and $\Vert \mathbf{y}\Vert$ with respect to this inner product.

Given

determine the value of each of the following:

1. $\langle A,B\rangle$

2. ${\Vert A\Vert }_F$

3. ${\Vert B\Vert }_F$

4. ${\Vert A+B\Vert }_F$

Show that equation (2) defines an inner product on ${ℝ}^{m\times n}$.

Show that the inner product defined by equation (3) satisfies the last two conditions of the definition of an inner product.

In C[0, 1], with inner product defined by (3), compute

1. $\langle e^x,e^{-x}\rangle$

2. $\langle x,\sin \pi x\rangle$

3. $\langle x^2,x^3\rangle$

In C[0, 1], with inner product defined by (3), consider the vectors 1 and x.

1. Find the angle $\theta$ between 1 and x.

2. Determine the vector projection p of 1 onto x and verify that $1-\mathbf{p}$ is orthogonal to p.

3. Compute $\Vert 1-\mathbf{p}\Vert ,\Vert \mathbf{p}\Vert ,\Vert 1\Vert$ and verify that the Pythagorean law holds.

In $C\left[-\pi ,\pi \right]$ with inner product defined by (6), show that cos mx and sin nx are orthogonal and that both are unit vectors. Determine the distance between the two vectors.

Show that the functions x and $x^2$ are orthogonal in $P_5$ with the inner product defined by (5), where $x_i=\left(i-3\right)/2$ for $i=1,\mathrm{\dots },5$.

In $P_5$ with the inner product as in Exercise 10 and the norm defined by

compute

1. $\Vert x\Vert$

2. $\Vert x^2\Vert$

3. the distance between x and $x^2$

If V is an inner product space, show that

satisfies the first two conditions in the definition of a norm.

Show that

defines a norm on ${ℝ}^n$.

Show that

defines a norm on ${ℝ}^n$.

Compute $\Vert \mathbf{x}{\Vert }_1,\Vert \mathbf{x}{\Vert }_2$ and $\Vert \mathbf{x}{\Vert }_{\infty }$ for each of the following vectors in ${ℝ}^3$:

1. $\mathbf{x}={\left(-3,4,0\right)}^T$

2. $\mathbf{x}={\left(-1,-1,2\right)}^T$

3. $\mathbf{x}={\left(1,1,1\right)}^T$

Let $\mathbf{x}={\left(5,2,4\right)}^T$ and $\mathbf{y}={\left(3,3,2\right)}^T$. Compute ${\Vert \mathbf{x}-\mathbf{y}\Vert }_1,{\Vert \mathbf{x}-\mathbf{y}\Vert }_1$, and ${\Vert \mathbf{x}-\mathbf{y}\Vert }_{\infty }$. Under which norm are the two vectors closest together? Under which norm are they farthest apart?

Let x and y be vectors in an inner product space. Show that if $\mathbf{x}\perp \mathbf{y}$, then the distance between x and y is

Show that if u and v are vectors in an inner product space that satisfy the Pythagorean law

then u and v must be orthogonal.

In ${ℝ}^n$ with inner product

derive a formula for the distance between two vectors $\mathbf{x}={\left(x_1,\mathrm{\dots },x_n\right)}^T$ and $\mathbf{y}={\left(y_1,\mathrm{\dots },y_n\right)}^T$.

Let A be a nonsingular $n\times n$ matrix and for each vector x in ${ℝ}^n$ define

Show that (11) defines a norm on ${ℝ}^n$.

Let $\mathbf{x}\in {ℝ}^n$. Show that ${\Vert \mathbf{x}\Vert }_{\infty }\le {\Vert \mathbf{x}\Vert }_2$.

Let $\mathbf{x}\in {ℝ}^2$. Show that ${\Vert \mathbf{x}\Vert }_2\le {\Vert \mathbf{x}\Vert }_1$. [Hint: Write x in the form $x_1{\mathbf{e}}_1+x_2{\mathbf{e}}_2$ and use the triangle inequality.]

Give an example of a nonzero vector $\mathbf{x}\in {ℝ}^2$ for which

Show that in any vector space with a norm,

Show that for any u and v in a normed vector space,

Prove that, for any u and v in an inner product space V,

Give a geometric interpretation of this result for the vector space ${ℝ}^2$.

The result of Exercise 26 is not valid for norms other than the norm derived from the inner product. Give an example of this in ${ℝ}^2$ using ${\Vert \cdot \Vert }_1$.

Determine whether the following define norms on C[a, b]:

1. $\Vert f\Vert =\left|f\left(a\right)\right|+\left|f\left(b\right)\right|$

2. $\Vert f\Vert ={\int }_a^b\left|f\left(x\right)\right|dx$

3. $\Vert f\Vert =\underset{a\le x\le b}{\max }\left|f\left(x\right)\right|$

Let x ∈ ${ℝ}^n$ and show that

1. ${\Vert \mathbf{x}\Vert }_1\le n{\Vert \mathbf{x}\Vert }_{\infty }$

2. ${\Vert \mathbf{x}\Vert }_2\le \sqrt{n}{\Vert \mathbf{x}\Vert }_{\infty }$

Give examples of vectors in ${ℝ}^n$ for which equality holds in parts (a) and (b).

Sketch the set of points $\left(x_1,x_2\right)={\mathbf{x}}^T$ in ${ℝ}^2$ such that

1. ${\Vert \mathbf{x}\Vert }_2=1$

2. ${\Vert \mathbf{x}\Vert }_1=1$

3. ${\Vert \mathbf{x}\Vert }_{\infty }=1$

Let K be an $n\times n$ matrix of the form

where $c^2+s^2=1$. Show that ${\Vert K\Vert }_F=\sqrt{n}$.

The trace of an $n\times n$ matrix C, denoted tr(C), is the sum of its diagonal entries; that is,

If A and B are $m\times n$ matrices, show that

1. ${\Vert A\Vert }_F^2=\text{tr}\left(A^{T}A\right)$

2. ${\Vert A+B\Vert }_F^2={\Vert A\Vert }_F^2+2\text{tr}\left(A^{T}B\right)+{\Vert B\Vert }_F^2$

Consider the vector space ${ℝ}^n$ with inner product $\langle \mathbf{x},\mathbf{y}\rangle ={\mathbf{x}}^T\mathbf{y}$. Show that for any $n\times n$ matrix A,

1. $\langle A\mathbf{x},\mathbf{y}\rangle =\langle \mathbf{x},A^T\mathbf{y}\rangle$

2. $\langle A^{T}A\mathbf{x},\mathbf{x}\rangle ={\left|\right|A\mathbf{x}\left|\right|}^2$