Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![## Vector Projection and Least Squares Solution in Linear Algebra
### Question 2
#### Part a)
Suppose \( H \) is a 2-dimensional subspace of \( \mathbb{R}^3 \) with a basis \( \{ \mathbf{b_1}, \mathbf{b_2} \} \), and \( \mathbf{y} \) is a vector given by
\[
\mathbf{b_1} = \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}, \quad \mathbf{b_2} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \quad \mathbf{y} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}
\]
Find \( \text{proj}_H \mathbf{y} \) and \( \mathbf{z} \) such that \( \mathbf{y} = \text{proj}_H \mathbf{y} + \mathbf{z} \).
**Hint:** Find the basis of \( H \), then use the inner product to find the coefficients.
#### Part b)
Find the least squares solution of the following equation and then find the least-squares error,
\[
\begin{pmatrix} 1 & 1 \\ -1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}
\]
**Hint:** For the equation \( A\mathbf{y} = \mathbf{b} \), the least-square solution can be found by solving \( A^T A \hat{\mathbf{y}} = A^T \mathbf{b} \). The error is the norm of \( \mathbf{b} - A \hat{\mathbf{y}} \).](https://content.bartleby.com/qna-images/question/238584dd-2c47-447e-aaba-b9fce1ec1b5c/8f3331c8-421c-48cd-990b-a9aa426a8c4b/lq263g_processed.png)
Transcribed Image Text:## Vector Projection and Least Squares Solution in Linear Algebra
### Question 2
#### Part a)
Suppose \( H \) is a 2-dimensional subspace of \( \mathbb{R}^3 \) with a basis \( \{ \mathbf{b_1}, \mathbf{b_2} \} \), and \( \mathbf{y} \) is a vector given by
\[
\mathbf{b_1} = \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}, \quad \mathbf{b_2} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \quad \mathbf{y} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}
\]
Find \( \text{proj}_H \mathbf{y} \) and \( \mathbf{z} \) such that \( \mathbf{y} = \text{proj}_H \mathbf{y} + \mathbf{z} \).
**Hint:** Find the basis of \( H \), then use the inner product to find the coefficients.
#### Part b)
Find the least squares solution of the following equation and then find the least-squares error,
\[
\begin{pmatrix} 1 & 1 \\ -1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}
\]
**Hint:** For the equation \( A\mathbf{y} = \mathbf{b} \), the least-square solution can be found by solving \( A^T A \hat{\mathbf{y}} = A^T \mathbf{b} \). The error is the norm of \( \mathbf{b} - A \hat{\mathbf{y}} \).
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