Solution Found!
Solved: In Exercises 17 and 18, all vectors and subspaces
Chapter 6, Problem 18E(choose chapter or problem)
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 = 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 \(\left\{\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}\right\}\) is an orthogonal set in W , then \(\left\{\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}\right\}\) is a basis for W .
b. If 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 = QR (when A has linearly independent columns), the columns of Q form an orthonormal basis for the column space of A.
Questions & Answers
QUESTION:
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 = 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 \(\left\{\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}\right\}\) is an orthogonal set in W , then \(\left\{\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}\right\}\) is a basis for W .
b. If 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 = QR (when A has linearly independent columns), the columns of Q form an orthonormal basis for the column space of A.
ANSWER:Solution 18E1. {}is a basis for W. Hence,