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Q17E

Expert-verifiedFound in: Page 331

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 independent**

**set \(\left\{ {{{\bf{x}}_1}, \ldots ,{{\bf{x}}_p}} \right\}\)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.

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 \({Q^T}A = R\).

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.

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