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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q26E (page 217)

Find the orthogonal projection of onto the subspace of spanned by and.

Short Answer

Expert verified

werrer

Step by step solution

01

Determine the orthonormal basis

Consider a set spanned and a vector .

If the vectoris perpendicularthen.

By the theorem of orthogonal basis, the orthogonal is defined as follows.

Substitute the value for in the equation as follows.

$\begin{aligned} {\overset{鈫�}{u}}_{1} = \frac{{\overset{鈫�}{v}}_{1}}{鈭�{\overset{鈫�}{v}}_{1}鈭�} \\ {\overset{鈫�}{u}}_{1} = \frac{[\begin{matrix} 2 \\ 3 \\ 6 \end{matrix}]}{鈭�[\begin{matrix} 2 \\ 3 \\ 6 \end{matrix}]鈭�} {\overset{鈫�}{u}}_{1} = \frac{1}{\sqrt{2^{2} + 3^{2} + 6^{2}}} [\begin{matrix} 3 \\ 6 \end{matrix}] \\ {\overset{鈫�}{u}}_{1} = \frac{1}{7} [\begin{matrix} 2 \\ 3 \\ 6 \end{matrix}] \end{aligned}$

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Most popular questions from this chapter

Consider an invertible n脳nmatrix A. Can you write Aas A=LQ, where Lis a lowertriangular matrix andQis orthogonal? Hint: Consider the QRfactorizationof $A^{T}$ .

Find scalar $a, b, c, d, e, f, g$ such that vectors $[\begin{matrix} a \\ d \\ f \end{matrix}], [\begin{matrix} b \\ 1 \\ g \end{matrix}], [\begin{matrix} c \\ \begin{matrix} e \\ \frac{1}{2} \end{matrix} \end{matrix}]$ are orthonormal.

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

$4 路 [\begin{matrix} 4 \\ 0 \\ 3 \end{matrix}], [\begin{matrix} 25 \\ 0 \\ - 25 \end{matrix}], [\begin{matrix} 0 \\ - 2 \\ 0 \end{matrix}]$

Question:TRUE OR FALSE?If matrices A and Bare commute, then A must commute $B^{T}$ withas well.

Consider an invertible n脳nmatrix A. Can you write A=RQ, where Ris an upper triangular matrix and Q is orthogonal?

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