Linear Algebra and Its Applications

David C. Lay, Steven R. Lay, Judi J. McDonald

Chapter 6

Orthogonality and Least Square - all with Video Answers

Educators

Section 4

The Gram–Schmidt Process

Problem 1

In Excrcises $1-6,$ the given set is a basis for a subspace $W .$ Use the Gram-Schmidt process to produce an orthogonal basis for $W$ .
$$
\left[\begin{array}{r}{3} \\ {0} \\ {-1}\end{array}\right],\left[\begin{array}{r}{8} \\ {5} \\ {-6}\end{array}\right]
$$

Ahmad R.

Ahmad R.

Numerade Educator

Problem 2

In Excrcises $1-6,$ the given set is a basis for a subspace $W .$ Use the Gram-Schmidt process to produce an orthogonal basis for $W$ .
$$
\left[\begin{array}{l}{0} \\ {4} \\ {2}\end{array}\right],\left[\begin{array}{r}{5} \\ {6} \\ {-7}\end{array}\right]
$$

Sean T.

Numerade Educator

Problem 3

In Excrcises $1-6,$ the given set is a basis for a subspace $W .$ Use the Gram-Schmidt process to produce an orthogonal basis for $W$ .
$$
\left[\begin{array}{r}{2} \\ {-5} \\ {1}\end{array}\right],\left[\begin{array}{r}{4} \\ {-1} \\ {2}\end{array}\right]
$$

Ahmad R.

Numerade Educator

Problem 4

In Excrcises $1-6,$ the given set is a basis for a subspace $W .$ Use the Gram-Schmidt process to produce an orthogonal basis for $W$ .
$$
\left[\begin{array}{r}{3} \\ {-4} \\ {5}\end{array}\right],\left[\begin{array}{c}{-3} \\ {14} \\ {-7}\end{array}\right]
$$

Sean T.

Numerade Educator

Problem 5

In Excrcises $1-6,$ the given set is a basis for a subspace $W .$ Use the Gram-Schmidt process to produce an orthogonal basis for $W$ .
$$
\left[\begin{array}{r}{1} \\ {-4} \\ {0} \\ {1}\end{array}\right],\left[\begin{array}{r}{7} \\ {-7} \\ {-4} \\ {1}\end{array}\right]
$$

Ahmad R.

Numerade Educator

Problem 6

In Excrcises $1-6,$ the given set is a basis for a subspace $W .$ Use the Gram-Schmidt process to produce an orthogonal basis for $W$ .
$$
\left[\begin{array}{r}{3} \\ {-1} \\ {2} \\ {-1}\end{array}\right],\left[\begin{array}{r}{-5} \\ {9} \\ {-9} \\ {3}\end{array}\right]
$$

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Problem 7

Find an orthonormal basis of the subspace spanned by the vectors in Exercise $3 .$

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Problem 8

Find an orthonormal basis of the subspace spanned by the vectors in Exercise $4 .$

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Problem 9

Find an orthogonal basis for the column space of each matrix in Exercises $9-12$ .
$$
\left[\begin{array}{rrr}{3} & {-5} & {1} \\ {1} & {1} & {1} \\ {-1} & {5} & {-2} \\ {3} & {-7} & {8}\end{array}\right]
$$

Ahmad R.

Numerade Educator

Problem 10

Find an orthogonal basis for the column space of each matrix in Exercises $9-12$ .
$$
\left[\begin{array}{rrr}{-1} & {6} & {6} \\ {3} & {-8} & {3} \\ {1} & {-2} & {6} \\ {1} & {-4} & {-3}\end{array}\right]
$$

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Problem 11

Find an orthogonal basis for the column space of each matrix in Exercises $9-12$ .
$$
\left[\begin{array}{rrr}{1} & {2} & {5} \\ {-1} & {1} & {-4} \\ {-1} & {4} & {-3} \\ {1} & {-4} & {7} \\ {1} & {2} & {1}\end{array}\right]
$$

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Problem 12

Find an orthogonal basis for the column space of each matrix in Exercises $9-12$ .
$$
\left[\begin{array}{rrr}{1} & {3} & {5} \\ {-1} & {-3} & {1} \\ {0} & {2} & {3} \\ {1} & {5} & {2} \\ {1} & {5} & {8}\end{array}\right]
$$

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Problem 13

In Exercises 13 and $14,$ the columns of $Q$ were obtained by applying the Gram-Schmidt process to the columns of $A .$ Find an upper triangular matrix $R$ such that $A=Q R .$ Check your work.
$$
A=\left[\begin{array}{rr}{5} & {9} \\ {1} & {7} \\ {-3} & {-5} \\ {1} & {5}\end{array}\right], Q=\left[\begin{array}{rr}{5 / 6} & {-1 / 6} \\ {1 / 6} & {5 / 6} \\ {-3 / 6} & {1 / 6} \\ {1 / 6} & {3 / 6}\end{array}\right]
$$

Ahmad R.

Numerade Educator

Problem 14

In Exercises 13 and $14,$ the columns of $Q$ were obtained by applying the Gram-Schmidt process to the columns of $A .$ Find an upper triangular matrix $R$ such that $A=Q R .$ Check your work.
$$
A=\left[\begin{array}{rr}{-2} & {3} \\ {5} & {7} \\ {2} & {-2} \\ {4} & {6}\end{array}\right], Q=\left[\begin{array}{rr}{-2 / 7} & {5 / 7} \\ {5 / 7} & {2 / 7} \\ {2 / 7} & {-4 / 7} \\ {4 / 7} & {2 / 7}\end{array}\right]
$$

Sean T.

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Problem 15

Find a QR factorization of the matrix in Exercise 11

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Problem 16

Find a QR factorization of the matrix in Exercise 12

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Problem 17

In Exercises 17 and $18,$ all vectors and subspaces are in $\mathbb{R}^{n} .$ Mark each statement True or False. Justify each answer.
a. If $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is an orthogonal basis for $W,$ then mul- tiplying $\mathbf{v}_{3}$ by a scalar $c$ gives a new orthogonal basis $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, c \mathbf{v}_{3}\right\} .$
b. The Gram-Schmidt process produces from a linearly in-
dependent set $\left\{\mathbf{x}_{1}, \ldots, \mathbf{x}_{p}\right\}$ an orthogonal set $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\right\}$ with the property that for each $k,$ the vectors $\mathbf{v}_{1}, \ldots, \mathbf{v}_{k}$ span the same subspace as that spanned by $\mathbf{x}_{1}, \ldots, \mathbf{x}_{k}$
c. If $A=Q R,$ where $Q$ has orthonormal columns, then $R=Q^{T} A$

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Problem 18

In Exercises 17 and $18,$ all vectors and subspaces are in $\mathbb{R}^{n} .$ Mark each statement True or False. Justify each answer.
a. If $W=\operatorname{Span}\left\{\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}\right\}$ with $\left\{\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}\right\}$ linearly independent, and if $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is an orthogonal set in $W,$ then
$\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is a basis for $W$
b. If $\mathbf{x}$ is not in a subspace $W,$ then $\mathbf{x}-\operatorname{proj}_{W} \mathbf{x}$ is not zero.
c. In a QR factorization, say $A=Q R$ (when $A$ has lincarly independent columns), the columns of $Q$ form an orthonormal basis for the column space of $A .$

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Problem 19

Suppose $A=Q R,$ where $Q$ is $m \times n$ and $R$ is $n \times n .$ Show that if the columns of $A$ are linearly independent, then $R$ must be invertible. $[\text { Hint: Study the equation } R \mathbf{x}=\mathbf{0} \text { and use the }$
fact that $A=Q R .$

Ahmad R.

Numerade Educator

Problem 20

Suppose $A=Q R,$ where $R$ is an invertible matrix. Show that $A$ and $Q$ have the same column space. IHint: Given $y$ in Col $A,$ show that $\mathbf{y}=Q \mathbf{x}$ for some $\mathbf{x} .$ Also, given $\mathbf{y}$ in $\operatorname{Col} Q$

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Problem 21

Given $A=Q R$ as in Theorem $12,$ describe how to find an orthogonal $m \times m$ (square) matrix $Q_{1}$ and an invertible $n \times n$ upper triangular matrix $R$ such that $A=Q_{1}\left[\begin{array}{l}{R} \\ {0}\end{array}\right]$

Ahmad R.

Numerade Educator

Problem 22

Let $\mathbf{u}_{1}, \ldots, \mathbf{u}_{p}$ be an orthogonal basis for a subspace $W$ of $\mathbb{R}^{n},$ and let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be defined by $T(\mathbf{x})=\operatorname{proj}_{W} \mathbf{x}$ Show that $T$ is a linear transformation.

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Problem 23

Suppose $A=Q R$ is a QR factorization of an $m \times n$ ma- trix $A$ (with linearly independent columns). Partition $A$ as $\left[A_{1} \quad A_{2}\right],$ where $A_{1}$ has $p$ columns. Show how to obtain a Q. factorization of $A_{1},$ and explain why your factorization has the appropriate properties.

Ahmad R.

Numerade Educator

Problem 24

[M] Use the Gram-Schmidt process as in Example 2 to produce an orthogonal basis for the column space of $$
A=\left[\begin{array}{rrrr}{-10} & {13} & {7} & {-11} \\ {2} & {1} & {-5} & {3} \\ {-6} & {3} & {13} & {-3} \\ {16} & {-16} & {-2} & {5} \\ {2} & {1} & {-5} & {-7}\end{array}\right]
$$

Sean T.

Numerade Educator

Problem 25

[MJ Use the method in this section to produce a QR factorization of the matrix in Exercise $24 .$

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Numerade Educator

Problem 26

[M] For a matrix program, the Gram-Schmidt process works better with orthonormal vectors. Starting with $\mathbf{x}_{1}, \ldots, \mathbf{x}_{p}$ as in Theorem $11,$ let $A=\left[\mathbf{x}_{1} \quad \cdots \quad x_{p}\right] .$ Suppose $Q$ is an $n \times k$ matrix whose columns form an orthonormal basis for the subspace $W_{k}$ spanned by the first $k$ columns of $A .$ Then for $\mathbf{x}$ in $\mathbb{R}^{n}, Q Q^{T} \mathbf{x}$ is the orthogonal projection of $\mathbf{x}$ onto $W_{k}$ (Theorem 10 in Section 6.3$) .$ If $\mathbf{x}_{k+1}$ is the next column of $A$ then equation $(2)$ in the proof of Theorem 11 becomes
$$
\mathbf{v}_{k+1}=\mathbf{x}_{k+1}-Q\left(Q^{T} \mathbf{x}_{k+1}\right)
$$

Ahmad R.

Numerade Educator