Chapter 6, Problem 1SE

(a)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is False.

Explanation of Solution

Given information:

The following statements refer to vectors in ${ℝ}^n\left(\text{or}{ℝ}^m\right)$ with the standard inner product.

Given statement:

The length of every vector is a positive number.

Explanation:

The length of the zero vector is zero.

Therefore, there is no necessary that the length of every vector is a positive number.

Hence, the given statement is False.

(b)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is True.

Explanation of Solution

Given statement:

A vector v and its negative, $-v$, have equal lengths.

Explanation:

For any scalar c, the length of cv is $\left|c\right|$ times the length of v.

$\Vert cv\Vert =\left|c\right|\Vert v\Vert$ (1)

Consider a vector x and scalar $c=-1$, refer to Equation (1).

$\begin{array}{c}\Vert -x\Vert =\Vert \left(-1\right)x\Vert \\ =\left|-1\right|\Vert x\Vert \\ =\Vert x\Vert \end{array}$

Therefore, a vector v and its negative, $-v$, have equal lengths.

Hence, the given statement is True.

(c)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is True.

Explanation of Solution

Given statement:

The distance between u and v is $\Vert u-v\Vert$.

Explanation:

Definition of Distance:

For u and v in ${ℝ}^n$, the distance between u and v, written as $\text{dist}\left(u,v\right)$, is the length of the vector $u-v$. That is,

$\text{dist}\left(u,v\right)=\Vert u-v\Vert$

Refer to the definition of distance, the given statement is correct.

Hence, the given statement is True.

(d)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is False.

Explanation of Solution

Given statement:

If r is any scalar, then $\Vert rv\Vert =r\Vert v\Vert$.

Explanation:

The correct explanation for the given statement is:

$\Vert rv\Vert =\left|r\right|\Vert v\Vert$

Therefore, the given statement is False.

(e)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is False.

Explanation of Solution

Given statement:

If two vectors are orthogonal, they are linearly independent.

Explanation:

For the vectors to be linearly independent, the two orthogonal vectors should be nonzero.

Therefore, the given statement is False.

(f)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is True.

Explanation of Solution

Given statement:

If x is orthogonal to both u and v, then x must be orthogonal to $u-v$.

Explanation:

Prove the following condition.

Consider $x\cdot u=0\text{and}x\cdot v=0$.

Then,

$\begin{array}{c}x\cdot \left(u-v\right)=x\cdot u-x\cdot v\\ =0\end{array}$

Therefore, given statement is true.

(g)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is True.

Explanation of Solution

Given statement:

If ${\Vert u+v\Vert }^2={\Vert u\Vert }^2+{\Vert v\Vert }^2$, then u and v are orthogonal.

Explanation:

Theorem 2 (Section 6.1):

The Pythagorean Theorem:

Two vectors u and v are orthogonal if and only if ${\Vert u+v\Vert }^2={\Vert u\Vert }^2+{\Vert v\Vert }^2$.

Refer to the Pythagorean Theorem, the given statement is true.

(h)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is True.

Explanation of Solution

Given statement:

If ${\Vert u-v\Vert }^2={\Vert u\Vert }^2+{\Vert v\Vert }^2$, then u and v are orthogonal.

Explanation:

Theorem 2 (Section 6.1):

The Pythagorean Theorem:

Two vectors u and v are orthogonal if and only if ${\Vert u+v\Vert }^2={\Vert u\Vert }^2+{\Vert v\Vert }^2$.

Replace $-v$ for v in the Theorem.

The value of ${\Vert -v\Vert }^2={\Vert v\Vert }^2$.

Refer to the Pythagorean Theorem, the given statement is true.

(i)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is False.

Explanation of Solution

Given statement:

The orthogonal projection of y onto u is a scalar multiple of y.

Explanation:

The orthogonal projection of y onto u is a scalar multiple of u, not y.

Therefore, the given statement is False.

(j)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is True.

Explanation of Solution

Given statement:

If a vector y coincides with its orthogonal projection onto a subspace W, then y is in W.

Explanation:

The orthogonal projection of any vector y onto W is always a vector in W.

Therefore, the given statement is true.

(k)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is True.

Explanation of Solution

Given statement:

The set of all vectors in ${ℝ}^n$ orthogonal to one fixed vector is a subspace of ${ℝ}^n$.

Explanation:

Consider these statements:

1. 1. A vector x is in $W^{\perp }$ if and only if x is orthogonal to every vector in a set that spans W.
2. 2. $W^{\perp }$ is a subspace of ${ℝ}^n$

Refer to these two statements the given statement is true.

(l)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is False.

Explanation of Solution

Given statement:

If W is a subspace of ${ℝ}^n$, then W and $W^{\perp }$ have no vectors in common.

Explanation:

W and $W^{\perp }$ have zero vectors in common.

Therefore, the given statement is False.

(m)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is True.

Explanation of Solution

Given statement:

If $\left\{v_1,v_2,v_3\right\}$ is an orthogonal set and if $c_1,c_2,\text{and}{c}_3$ are scalars, then $\left\{c_1{v}_1,c_2{v}_2,c_3{v}_3\right\}$ is an orthogonal set.

Explanation:

Refer Exercise 32 in the section 6.2:

Consider $v_i\cdot v_j=0$.

Then,

$\begin{array}{c}\left(c_i{v}_i\right)\cdot \left(c_j{v}_j\right)=c_i{c}_j\left(v_i\cdot v_j\right)\\ =c_i{c}_j\left(0\right)\\ =0\end{array}$

Therefore, the given statement is true.

(n)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is False.

Explanation of Solution

Given statement:

If a matrix U has orthonormal columns, then $U{U}^T=I$.

Explanation:

Theorem 10 (Section 6.3):

If $\left\{u_1,\dots ,u_p\right\}$ is an orthonormal basis for a subspace Wof ${ℝ}^n$, then

${\text{proj}}_{W}y=\left(y\cdot u_1\right)u_1+\left(y\cdot u_2\right)u_2+\cdots +\left(y\cdot u_p\right)u_p$

If $U=\left[\begin{array}{cccc}{u}_1& u_2& \cdots & u_p\end{array}\right]$, then

${\text{proj}}_{W}y=U{U}^{T}y$ for all y in ${ℝ}^n$.

Refer to Theorem 10:

The statement is true only for square matrix.

Therefore, the given statement is False.

(o)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is False.

Explanation of Solution

Given statement:

A square matrix with orthogonal columns is an orthogonal matrix.

Explanation:

The statement should be an orthogonal matrix is square and has orthogonal columns.

Therefore, the given statement is False.

(p)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is True.

Explanation of Solution

Given statement:

If a square matrix has orthonormal columns, then it also has orthonormal rows.

Explanation:

Consider U has orthonormal columns.

Then,

$U^{T}U=I$.

Consider U is also square matrix.

Refer to the Invertible Matrix Theorem;

U is invertible and $U^{-1}=U^T$.

Here, $U^{T}U=I$ indicates that the column of $U^T$ are orthonormal; that is the rows of the matrix U are orthonormal.

Therefore, the given statement is true.

(q)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is True.

Explanation of Solution

Given statement:

If W is a subspace, then ${\Vert {\text{proj}}_{W}v\Vert }^2+{\Vert v-{\text{proj}}_{w}v\Vert }^2={\Vert v\Vert }^2$.

Explanation:

Theorem 2: The Pythagorean Theorem: (Chapter 6.1):

Two vectors u and v are orthogonal if and only if ${\Vert u+v\Vert }^2={\Vert u\Vert }^2+{\Vert v\Vert }^2$.

Theorem 8: The orthogonal Decomposition Theorem: (Chapter 6.3):

Let W be a subspace of ${ℝ}^n$. Then each y in ${ℝ}^n$ can be written uniquely in the form

$y=\widehat{y}+z$

Where $\widehat{y}$ is in W and z is in $W^{\perp }$. In fact, if $\left\{u_1,\dots ,u_p\right\}$ is any orthogonal basis of W, then

$\widehat{y}=\frac{y\cdot u_1}{u_1\cdot u_1}{u}_1+\cdots +\frac{y\cdot u_p}{u_p\cdot u_p}{u}_p$

and $z=y-\widehat{y}$.

Refer to Theorem 8; the vectors ${\text{proj}}_{W}v$ and $-{\text{proj}}_{W}v$ are orthogonal.

Refer to Equation 2; the given statement follows theorem 2.

Therefore, the given statement is true.

(r)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is False.

Explanation of Solution

Given statement:

A least-square solution of $Ax=b$ is the vector $A\widehat{x}$ in Col A closest to b, so that $\Vert b-A\widehat{x}\Vert \le \Vert b-Ax\Vert$ for all x.

Explanation:

A least-square solution of a vector is $\widehat{x}$ (not $A\widehat{x}$) likewise $A\widehat{x}$ is the closest point to b in Col A.

Therefore, the given statement is False.

(s)

To determine

To mark: each statement True or False.

To justify: Each answer.

Answer to Problem 1SE

The given statement is False.

Explanation of Solution

Given statement:

The normal equations for a least-squares solution of $Ax=b$ are given by $\widehat{x}={\left(A^{T}A\right)}^{-1}{A}^{T}b$.

Explanation:

The equation $\widehat{x}={\left(A^{T}A\right)}^{-1}{A}^{T}b$ describes the solution of the normal equations, not the matrix form of normal equations.

Therefore, the given statement is False.