Q67E Consider a subspace V of 鈻聽w... [FREE SOLUTION]

Chapter 5: Q67E (page 235)

Consider a subspace V of ${鈻�}^{n}$ with a basis ${\overset{鈬赌}{v}}_{1}, {\overset{鈬赌}{v}}_{2}, . . ., {\overset{鈬赌}{v}}_{m}$ ; suppose we wish to find a formula for the orthogonal projection onto V. Using the methods we have developed thus far, we can proceed in two steps: We use the Gram-Schmidt process to construct an orthonormal basis ${\overset{鈬赌}{u}}_{1}, . . ., {\overset{鈬赌}{u}}_{m}$ of V, and then we use Theorem 5.3.10: The matrix of the orthogonal projection QQ^T, where $Q = [{\overset{鈬赌}{u}}_{1} . . . {\overset{鈬赌}{u}}_{m}]$ . In this exercise we will see how we write the matrix of the projection directly in terms of the basis ${\overset{鈬赌}{v}}_{1}, . . ., {\overset{鈬赌}{v}}_{m}$ and the matrix $A = [{\overset{鈬赌}{v}}_{1} . . . {\overset{鈬赌}{v}}_{m}]$ . (This issue will be discussed more thoroughly in Section 5.4: see theorem 5.4.7.)
Since $p r o j_{v} \overset{鈬赌}{x}$ is in V, we can write $p r o j_{v} \overset{鈬赌}{x} = c, {\overset{鈬赌}{v}}_{1} + . . . + c_{m} {\overset{鈬赌}{v}}_{m}$ for some scalars $c_{1}, . . ., c_{m}$ yet to be determined. Now $\overset{鈬赌}{x} - p r o j_{v} (\overset{鈬赌}{x}) = \overset{鈬赌}{x} - c_{1} {\overset{鈬赌}{v}}_{1} - . . . - c_{m} {\overset{鈬赌}{v}}_{m}$ is orthogonal to V, meaning that role="math" localid="1660623171035" ${\overset{鈬赌}{v}}_{i} 路 (\overset{鈬赌}{x} - c_{1} {\overset{鈬赌}{v}}_{1} - . . . - c_{m} {\overset{鈬赌}{v}}_{m}) = 0$ for $i = 1, . . ., m$ .
Use the equation ${\overset{鈬赌}{v}}_{i} 路 (\overset{鈬赌}{x} - c_{1} {\overset{鈬赌}{v}}_{1} - . . . - c_{m} {\overset{鈬赌}{v}}_{m}) = 0$ to show that $A^{T} A \overset{鈬赌}{c} = A^{T} \overset{鈬赌}{x}$ , where $\overset{鈬赌}{c} = [\begin{matrix} c_{1} \\ 鈰� \\ c_{m} \end{matrix}]$ .
Conclude that $\overset{鈬赌}{c} = {(A^{T} A)}^{- 1} A^{T} \overset{鈬赌}{x}$ and $p r o j_{v} \overset{鈬赌}{x} = A \overset{鈬赌}{c} = A {(A^{T} A)}^{- 1} A^{T} \overset{鈬赌}{x}$ .

Short Answer

Expert verified

It is proved that $A^{T} A \overset{鈬赌}{c} = A^{T} \overset{鈬赌}{x}$ using the equation ${\overset{鈬赌}{v}}_{i} 路 (x - c_{1} {\overset{鈬赌}{v}}_{1} - . . . - c_{m} {\overset{鈬赌}{v}}_{m}) = 0$ .
It is concluded thatrole="math" localid="1660624017029" $\overset{鈬赌}{c} = {(A^{T} A)}^{- 1} A^{T} \overset{鈬赌}{x}$ and role="math" localid="1660626446185" $p r o j_{v} \overset{鈬赌}{x} = A \overset{鈬赌}{c} = A {(A^{T} A)}^{- 1} A^{T} \overset{鈬赌}{x}$ .

Step by step solution

Determine Orthogonal matrix

An $n 脳 n$ matrix $A$ is considered orthogonal if and only if $A^{T} A = l_{n}$ or we can say $A^{- 1} = A^{T}$ .

Determine proof of part (a)

Given that $\overset{鈬赌}{c} = [\begin{matrix} c_{1} \\ 鈰� \\ c_{m} \end{matrix}]$ . Let a matrix A be $A = [{\overset{鈬赌}{v}}_{1} . . . {\overset{鈬赌}{v}}_{m}]$ . So, the value of $A^{T} A \overset{鈬赌}{c}$ can be obtained as follows:

$\begin{array}{rcl} A^{T} A \overset{鈬赌}{c} & = & A^{T} [{\overset{鈬赌}{v}}_{1} . . . {\overset{鈬赌}{v}}_{m}] [\begin{matrix} c_{1} \\ 鈰� \\ c_{m} \end{matrix}] \\ = & A^{T} [c_{1} {\overset{鈬赌}{v}}_{1} . . . c_{m} {\overset{鈬赌}{v}}_{m}] \end{array}$

Now, we have role="math" localid="1660625187075" $\begin{array}{rcl} {\overset{鈬赌}{v}}_{i} 路 (\overset{鈬赌}{x} - c_{1} {\overset{鈬赌}{v}}_{1} - . . . - c_{m} {\overset{鈬赌}{v}}_{m}) & = & 0 \end{array}$ . This can be manipulated as:

$\begin{array}{rcl} {\overset{鈬赌}{v}}_{i} 路 (\overset{鈬赌}{x} - c_{1} {\overset{鈬赌}{v}}_{1} - . . . - c_{m} {\overset{鈬赌}{v}}_{m}) & = & 0 \\ (\overset{鈬赌}{x} - c_{1} {\overset{鈬赌}{v}}_{1} - . . . - c_{m} {\overset{鈬赌}{v}}_{m}) & = & 0 \\ \overset{鈬赌}{x} & = & c_{1} {\overset{鈬赌}{v}}_{1} + . . . + c_{m} {\overset{鈬赌}{v}}_{m} \end{array}$

So, the value of $A^{T} A \overset{鈬赌}{c}$ will become:

$\begin{array}{rcl} A^{T} A \overset{鈬赌}{c} & = & A^{T} [c_{1} {\overset{鈬赌}{v}}_{1} . . . c_{m} {\overset{鈬赌}{v}}_{m}] \\ = & A^{T} \overset{鈬赌}{x} \end{array}$

Thus, it is proved that $A^{T} A \overset{鈬赌}{c} = A^{T} \overset{鈬赌}{x}$ using the equation $\begin{array}{rcl} {\overset{鈬赌}{v}}_{i} 路 (\overset{鈬赌}{x} - c_{1} {\overset{鈬赌}{v}}_{1} - . . . - c_{m} {\overset{鈬赌}{v}}_{m}) & = & 0 \end{array}$ .

Determine the proof of part (b)

From part (a), we have $A^{T} A \overset{鈬赌}{c} = A^{T} \overset{鈬赌}{x}$ . On multiplying ${(A^{T} A)}^{- 1}$ both sides to the equation, we get role="math" localid="1660625828176" ${(A^{T} A)}^{- 1} (A^{T} A \overset{鈬赌}{c}) = {(A^{T} A)}^{- 1} (A^{T} \overset{鈬赌}{x})$ . Simplify the equation as follows:

$\begin{array}{rcl} {(A^{T} A)}^{- 1} (A^{T} A \overset{鈬赌}{c}) & = & {(A^{T} A)}^{- 1} (A^{T} \overset{鈬赌}{x}) \\ {(A^{T} A)}^{- 1} (A^{T} A) (\overset{鈬赌}{c}) & = & {(A^{T} A)}^{- 1} (A^{T} \overset{鈬赌}{x}) \\ \overset{鈬赌}{c} & = & {(A^{T} A)}^{- 1} (A^{T} \overset{鈬赌}{x}) \end{array}$

It is known that role="math" localid="1660626003168" $p r o j_{v} \overset{鈬赌}{x} = c_{1} {\overset{鈬赌}{v}}_{1} + . . . + c_{m} {\overset{鈬赌}{v}}_{m}$ . It can be manipulated as:

$\begin{array}{rcl} p r o j_{v} \overset{鈬赌}{x} & = & c_{1} {\overset{鈬赌}{v}}_{1} + . . . + c_{m} {\overset{鈬赌}{v}}_{m} \\ = & [{\overset{鈬赌}{v}}_{1} . . . {\overset{鈬赌}{v}}_{m}] [\begin{matrix} c_{1} \\ 鈰� \\ c_{m} \end{matrix}] \\ = & A \overset{鈬赌}{c} \end{array}$

Now, we have the value ${(A^{T} A)}^{- 1} (A^{T} \overset{鈬赌}{x})$ . Put this value in the above equation to get:

$\begin{array}{rcl} p r o j_{v} \overset{鈬赌}{x} & = & A \overset{鈬赌}{c} \\ = & A ({(A^{T} A)}^{- 1} (A^{T} \overset{鈬赌}{x})) \\ = & A {(A^{T} A)}^{- 1} (A^{T} \overset{鈬赌}{x}) \end{array}$

Thus, it is concluded that $\overset{鈬赌}{c} = {(A^{T} A)}^{- 1} A^{T} \overset{鈬赌}{x}$ and $p r o j_{v} \overset{鈬赌}{x} = A \overset{鈬赌}{c} = A {(A^{T} A)}^{- 1} A^{T} \overset{鈬赌}{x}$ .

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Short Answer

Step by step solution

Determine Orthogonal matrix

Determine proof of part (a)

Determine the proof of part (b)

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