• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon

Suggested languages for you:

Americas

Europe

Q17E

Expert-verified
Found in: Page 331

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# In Exercises 17 and 18, all vectors and subspaces are in $${\mathbb{R}^n}$$. Mark each statement True or False. Justify each answer.17. a.If $$\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}$$ is an orthogonal basis for$$W$$, then multiplying$${v_3}$$ by a scalar $$c$$ gives a new orthogonal basis $$\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},c{{\bf{v}}_3}} \right\}$$.b. The Gram–Schmidt process produces from a linearly independentset $$\left\{ {{{\bf{x}}_1}, \ldots ,{{\bf{x}}_p}} \right\}$$an orthogonal set $$\left\{ {{{\bf{v}}_1}, \ldots ,{{\bf{v}}_p}} \right\}$$ with the property that for each $$k$$, the vectors $${{\bf{v}}_1}, \ldots ,{{\bf{v}}_k}$$ span the same subspace as that spanned by $${{\bf{x}}_1}, \ldots ,{{\bf{x}}_k}$$.c. If $$A = QR$$, where $$Q$$ has orthonormal columns, then $$R = {Q^T}A$$.

a. False, because this statement is true when $$c \ne 0$$.

b. True, because $${\rm{span}}\left\{ {{x_1}, \ldots ,{x_p}} \right\} = {\rm{span}}\left\{ {{v_1}, \ldots ,{v_p}} \right\}$$.

c. True, by using the definition of $$QR$$ factorization of a matrix.

See the step by step solution

## Step 1: $$QR$$ factorization of a Matrix

A matrix which has order $$m \times n$$ can be written as the multiplication of a upper triangular matrix $$R$$ and a matrix $$Q$$ which is formed by applying Gram–Schmidt orthogonalization process to the $${\rm{col}}\left( A \right)$$.

The matrix $$R$$ can be found by the formula $${Q^T}A = R$$.

## Step 2: Checking whether the given statements are true of false

a.

This is false. Because this statement is true when $$c \ne 0$$.

If $$c = 0$$, then we will have $$\left\{ {{v_1},{v_2},0} \right\}$$. We know that any set containing zero vector is not a linearly independent set.

b.

The given statement is true. From the Theorem 11, since $$\left\{ {{x_1}, \ldots ,{x_p}} \right\}$$gives the set of orthogonal vectors $$\left\{ {{v_1}, \ldots ,{v_p}} \right\}$$, thus we have

$${\rm{span}}\left\{ {{x_1}, \ldots ,{x_p}} \right\} = {\rm{span}}\left\{ {{v_1}, \ldots ,{v_p}} \right\}$$.

c.

The statement is true. From the definition of $$QR$$ factorization of a matrix it is true.

We have $${Q^T}Q = I$$, since the columns of the matrix $$Q$$ are orthonormal.

\begin{aligned}{}A &= QR\\{Q^T}A &= {Q^T}QR\\{Q^T}A &= IR\\{Q^T}A &= R\end{aligned}

Thus, the statement is true.