Let W be the subspace spanned by the u's. Note that the u's are pependicular. A) Find the orthogonal projection of y on W B) Find a vector orthogonal to W. C) Write y as the sum of a vector in W and a vector orthogonal to W.
Let W be the subspace spanned by the u's. Note that the u's are pependicular. A) Find the orthogonal projection of y on W B) Find a vector orthogonal to W. C) Write y as the sum of a vector in W and a vector orthogonal to W.
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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Let W be the subspace spanned by the u's. Note that the u's are pependicular.
A) Find the orthogonal projection of y on W
B) Find a
C) Write y as the sum of a vector in W and a vector orthogonal to W.
![The image shows three column vectors:
\[
\mathbf{y} = \begin{bmatrix} 24 \\ 4 \\ 10 \end{bmatrix}, \quad \mathbf{u}_1 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \quad \mathbf{u}_2 = \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix}
\]
- **Vector \(\mathbf{y}\)** has elements 24, 4, and 10.
- **Vector \(\mathbf{u}_1\)** has elements 1, 0, and -1.
- **Vector \(\mathbf{u}_2\)** has elements 2, 1, and 2.
These vectors might be part of a linear algebra context, such as vector spaces, linear transformations, or systems of equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa67aba50-2b96-4b32-9523-eddb477e2ff0%2Ff8df44ca-5b95-4bde-a706-426f8921e5be%2Fp3rkipb_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows three column vectors:
\[
\mathbf{y} = \begin{bmatrix} 24 \\ 4 \\ 10 \end{bmatrix}, \quad \mathbf{u}_1 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \quad \mathbf{u}_2 = \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix}
\]
- **Vector \(\mathbf{y}\)** has elements 24, 4, and 10.
- **Vector \(\mathbf{u}_1\)** has elements 1, 0, and -1.
- **Vector \(\mathbf{u}_2\)** has elements 2, 1, and 2.
These vectors might be part of a linear algebra context, such as vector spaces, linear transformations, or systems of equations.
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