Q69E In, consider the subspace W聽spa... [FREE SOLUTION]

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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: Q69E (page 235)

In, consider the subspace Wspanned by the vectors and $[\begin{matrix} 1 \\ 1 \\ - 1 \\ 0 \end{matrix}] and [\begin{matrix} 0 \\ 1 \\ 1 \\ - 1 \end{matrix}]$ . Find the matrixof $P_{w}$ the orthogonal projection onto W.

Short Answer

Expert verified

The matrix is $P_{w} = 1 / 3 [\begin{matrix} 1 & 1 & - 1 & 0 \\ 1 & 2 & 0 & - 1 \\ - 1 & 0 & 2 & - 1 \\ 0 & - 1 & - 1 & 1 \end{matrix}]$ .

Step by step solution

Matrix Pwof the orthogonal projection onto W.

Here is need to find the matrix $P_{w}$ of the orthogonal projection onto W.

So, according to the Exercise 67, the required matrix will be $A {(A^{T} A)}^{- 1} A^{T}$ , where the matrix .

$A = [{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}]$

Then, consider the equations below.

$A^{T} A = [\begin{matrix} 1 & 1 & - 1 & 0 \\ 0 & 1 & 1 & - 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ 1 & 1 \\ - 1 & 1 \\ 0 & - 1 \end{matrix}] = [\begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}]$

role="math" localid="1660131299371" $A^{T} {(A)}^{- 1} = 1 / 3 [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] A {(A^{T} A)}^{- 1} = 1 / 3 [\begin{matrix} 1 & 0 \\ 1 & 1 \\ - 1 & 1 \\ 0 & - 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] = 1 / 3 [\begin{matrix} 1 & 0 \\ 1 & 1 \\ - 1 & 1 \\ 0 & - 1 \end{matrix}]$

Now $A {(A^{T} A)}^{- 1} A^{T}$ is find as,

$A (A^{T} {A)}^{- 1} A^{T} = [\begin{matrix} 1 & 0 \\ 1 & 1 \\ - 1 & 1 \\ 0 & - 1 \end{matrix}] [\begin{matrix} 1 & 1 & - 1 & 0 \\ 0 & 1 & 1 & - 1 \end{matrix}] = 1 / 3 [\begin{matrix} 1 & 1 & - 1 & 0 \\ 1 & 2 & 0 & - 1 \\ - 1 & 0 & 2 & - 1 \\ 0 & - 1 & - 1 & 1 \end{matrix}]$

Hence, the required matrix is $P_{w} = 1 / 3 [\begin{matrix} 1 & 1 & - 1 & 0 \\ 1 & 2 & 0 & - 1 \\ - 1 & 0 & 2 & - 1 \\ 0 & - 1 & - 1 & 1 \end{matrix}]$ .

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91影视

In, consider the subspace Wspanned by the vectors and $[\begin{matrix} 1 \\ 1 \\ - 1 \\ 0 \end{matrix}] and [\begin{matrix} 0 \\ 1 \\ 1 \\ - 1 \end{matrix}]$ . Find the matrixof $P_{w}$ the orthogonal projection onto W.

Short Answer

Step by step solution

Matrix Pwof the orthogonal projection onto W.

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91影视

In, consider the subspace Wspanned by the vectors and [11-10]and[011-1]. Find the matrixof Pwthe orthogonal projection onto W.

Short Answer

Step by step solution

Matrix Pwof the orthogonal projection onto W.

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Most popular questions from this chapter

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Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

In, consider the subspace Wspanned by the vectors and $[\begin{matrix} 1 \\ 1 \\ - 1 \\ 0 \end{matrix}] and [\begin{matrix} 0 \\ 1 \\ 1 \\ - 1 \end{matrix}]$ . Find the matrixof $P_{w}$ the orthogonal projection onto W.