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1. If ✩ is the least squares solution of Ax = b, then Ax is the projection of b onto R(A). 2. Let V be an inner product space. Then (x, y) = (y,x) for all x and y in V. 3. If Y is a subspace of R", then dim(Y) + dim(Y¹) = n. False 4. The distance from a vector y to the line spanned by a vector v is given by y - vl. True False True

1. If ✩ is the least squares solution of Ax = b, then Ax is the projection of b onto R(A). 2. Let V be an inner product space. Then (x, y) = (y,x) for all x and y in V. 3. If Y is a subspace of R", then dim(Y) + dim(Y¹) = n. False 4. The distance from a vector y to the line spanned by a vector v is given by y - vl. True False True

Advanced Engineering Mathematics
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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### Linear Algebra: True or False Questions with Explanations #### Statement 1: **True** **Explanation:** If \( \hat{x} \) is the least squares solution of \( Ax = b \), then \( A\hat{x} \) is the projection of \( b \) onto \( \mathcal{R}(A) \). #### Statement 2: **False** **Explanation:** Let \( V \) be an inner product space. Then \( \langle x, y \rangle = \langle y, x \rangle \) for all \( x \) and \( y \) in \( V \). #### Statement 3: **True** **Explanation:** If \( Y \) is a subspace of \( \mathbb{R}^n \), then \( \text{dim}(Y) + \text{dim}(Y^\perp) = n \). #### Statement 4: **False** **Explanation:** The distance from a vector \( y \) to the line spanned by a vector \( v \) is given by \( \frac{\| y - v \|}{\|v\|} \).
Transcribed Image Text:### Linear Algebra: True or False Questions with Explanations #### Statement 1: **True** **Explanation:** If \( \hat{x} \) is the least squares solution of \( Ax = b \), then \( A\hat{x} \) is the projection of \( b \) onto \( \mathcal{R}(A) \). #### Statement 2: **False** **Explanation:** Let \( V \) be an inner product space. Then \( \langle x, y \rangle = \langle y, x \rangle \) for all \( x \) and \( y \) in \( V \). #### Statement 3: **True** **Explanation:** If \( Y \) is a subspace of \( \mathbb{R}^n \), then \( \text{dim}(Y) + \text{dim}(Y^\perp) = n \). #### Statement 4: **False** **Explanation:** The distance from a vector \( y \) to the line spanned by a vector \( v \) is given by \( \frac{\| y - v \|}{\|v\|} \).
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