Vi v₁ = 1. Let U span (G.D) be a subspace of R³. Answer the following questions based on this given U and the use of the dot product as the inner product: (a) Find a basis for U. basis -B (b) Let x = i. Using the basis you found in (a), create a matrix B and use this matrix to find the coordinate vector, A, of x in terms of subspace U. pointe) [ ] ii. Using your answer in (b), compute Tu(x), the orthogonal pro- jection of x onto the subspace U. pts)
Vi v₁ = 1. Let U span (G.D) be a subspace of R³. Answer the following questions based on this given U and the use of the dot product as the inner product: (a) Find a basis for U. basis -B (b) Let x = i. Using the basis you found in (a), create a matrix B and use this matrix to find the coordinate vector, A, of x in terms of subspace U. pointe) [ ] ii. Using your answer in (b), compute Tu(x), the orthogonal pro- jection of x onto the subspace U. pts)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 1: Subspace and Projection**
Let \( U = \text{span} \left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right\} \) be a subspace of \( \mathbb{R}^3 \). Answer the following questions based on this given \( U \) and the use of the dot product as the inner product.
(a) **Find a Basis for \( U \).**
- **Solution:** The basis for \( U \) is \( \left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right\} \).
(b) Let \( x = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \).
i. **Using the basis you found in (a), create a matrix \( B \) and use this matrix to find the coordinate vector, \(\lambda\), of \( x \) in terms of subspace \( U \).**
- **Solution:** Write the vector \( x \) as a linear combination of the basis vectors. Find the coefficients that satisfy:
\[
x = \lambda_1 \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + \lambda_2 \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}
\]
Solve for \(\lambda_1\) and \(\lambda_2\).
ii. **Using your answer in (b), compute \(\pi_U(x)\), the orthogonal projection of \( x \) onto the subspace \( U \).**
- **Solution:** Use the formula for orthogonal projection:
\[
\pi_U(x) = \sum_{i=1}^{2} \frac{\langle x, v_i \rangle}{\langle v_i, v_i \rangle} v_i
\]
Where \( v_1 \) and \( v_2 \) are the basis vectors.
This problem involves finding a basis for a given subspace, expressing a vector as a linear combination](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e25041d-7573-46df-b9d3-ec2dd7694c16%2F49ca124d-2aef-4a83-a604-b3caa5310560%2F44qvi7g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1: Subspace and Projection**
Let \( U = \text{span} \left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right\} \) be a subspace of \( \mathbb{R}^3 \). Answer the following questions based on this given \( U \) and the use of the dot product as the inner product.
(a) **Find a Basis for \( U \).**
- **Solution:** The basis for \( U \) is \( \left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right\} \).
(b) Let \( x = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \).
i. **Using the basis you found in (a), create a matrix \( B \) and use this matrix to find the coordinate vector, \(\lambda\), of \( x \) in terms of subspace \( U \).**
- **Solution:** Write the vector \( x \) as a linear combination of the basis vectors. Find the coefficients that satisfy:
\[
x = \lambda_1 \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + \lambda_2 \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}
\]
Solve for \(\lambda_1\) and \(\lambda_2\).
ii. **Using your answer in (b), compute \(\pi_U(x)\), the orthogonal projection of \( x \) onto the subspace \( U \).**
- **Solution:** Use the formula for orthogonal projection:
\[
\pi_U(x) = \sum_{i=1}^{2} \frac{\langle x, v_i \rangle}{\langle v_i, v_i \rangle} v_i
\]
Where \( v_1 \) and \( v_2 \) are the basis vectors.
This problem involves finding a basis for a given subspace, expressing a vector as a linear combination
Transcribed Image Text:(c) Using the Gram-Schmidt orthogonalization, turn the basis of \( U \) you got in (a) into an orthogonal basis.
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