Q39E Consider a line L聽in Rn, spanne... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q39E (page 233)

Consider a line Lin $R^{n}$ , spanned by a unit vector
$\overset{鈫�}{u} = [\begin{matrix} u_{1} \\ u_{2} \\ . \\ . \\ . \\ u_{n} \end{matrix}]$
Consider the matrix Aof the orthogonal projection onto L. Describe the ijth entry of A, in terms of the components ${\overset{鈫�}{u}}_{i}$ of $\overset{鈫�}{u}$ .

Short Answer

Expert verified

The ijth entry of A is given by ${\overset{鈫�}{u}}_{1} . {\overset{鈫�}{u}}_{j}$ .

Step by step solution

The matrix of an orthogonal projection.

The matrix P of the orthogonal projection onto V=spanis ${{\overset{鈫�}{u}}_{1} {\overset{鈫�}{u}}_{2}, . . . . . {\overset{鈫�}{u}}_{m}}$ given by, $P = Q Q^{T}$ where $Q = {{\overset{鈫�}{u}}_{1} {\overset{鈫�}{u}}_{2}, . . . . . {\overset{鈫�}{u}}_{m}}$ .

Where the spanning set is orthonormal basis.

Thus, in our case the vector subspace is L and therefore, the required projection matrix will be,

$A = Q Q^{T} = [\begin{matrix} u_{1} \\ u_{2} \\ . \\ . \\ . \\ u_{n} \end{matrix}] [\begin{matrix} u_{1} & u_{2} . . . . . u_{m} \end{matrix}] = [\begin{matrix} u_{1}^{2} & u_{2} u_{1} . . . . . . u_{1} u_{n} \\ u_{2} u_{1} & u_{1}^{2} . . . . . . . . . u_{2} u_{n} \\ . & . . . . . . . . . \\ . & . . . . . . . . . \\ . & . . . . . . . \\ u_{n} u_{1} & u_{2} u_{n} . . . . . . . u_{n} u_{n} \end{matrix}]$

Hence, the ijth entry of the matrix A is given by ${\overset{鈫�}{u}}_{i} {\overset{鈫�}{u}}_{j}$ .

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Short Answer

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The matrix of an orthogonal projection.

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91影视

Consider a line Lin Rn, spanned by a unit vectoru鈫�=[u1u2...un]Consider the matrix Aof the orthogonal projection onto L. Describe the ijth entry of A, in terms of the componentsu鈫�iof u鈫�.

Short Answer

Step by step solution

The matrix of an orthogonal projection.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Theoretical and Mathematical Physics

Statistics

Geometry

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Applied Mathematics

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