Let 0 -1 x = 2 -2 and A = 0 3 0 -2 -4 -4 0 3 0 -2 00 -4 0 1 -1 0 1 1 00 (a) Find a basis for Nul(A). (b) Find the projection of on to Nul(A). Call this vector *Nul. (c) Find a basis for Row(A). (d) Find the projection of x on to Row(A). Call this vector XRow. (e) Show that XNul and XRow are orthogonal. Will this be the case for all matrices A and all vectors x? Explain.
Let 0 -1 x = 2 -2 and A = 0 3 0 -2 -4 -4 0 3 0 -2 00 -4 0 1 -1 0 1 1 00 (a) Find a basis for Nul(A). (b) Find the projection of on to Nul(A). Call this vector *Nul. (c) Find a basis for Row(A). (d) Find the projection of x on to Row(A). Call this vector XRow. (e) Show that XNul and XRow are orthogonal. Will this be the case for all matrices A and all vectors x? Explain.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Linear Algebra Problem Set
Let
\[
x = \begin{bmatrix}
-1 \\
0 \\
2 \\
-2 \\
1
\end{bmatrix}
\quad \text{and} \quad
A = \begin{bmatrix}
3 & 0 & 3 & 0 & 0 \\
0 & -2 & -2 & 0 & 0 \\
0 & -4 & -4 & 0 & 1 \\
3 & -4 & -1 & 0 & 1 \\
1 & 0 & 1 & 0 & 0
\end{bmatrix}
\]
(a) **Find a basis for Nul(A).**
(b) **Find the projection of** \( \vec{x} \) **onto Nul(A). Call this vector** \( \vec{x}_{\text{Nul}} \).
(c) **Find a basis for Row(A).**
(d) **Find the projection of** \( \vec{x} \) **onto Row(A). Call this vector** \( \vec{x}_{\text{Row}} \).
(e) **Show that** \( \vec{x}_{\text{Nul}} \) **and** \( \vec{x}_{\text{Row}} \) **are orthogonal. Will this be the case for all matrices** \( A \) **and all vectors** \( \vec{x} \)? **Explain.**
---
### Explanation of the Diagram
The image provided is a linear algebra problem involving a vector \( x \) and a matrix \( A \). The tasks are to:
1. Determine a basis for the null space of \( A \), denoted as Nul(A).
2. Project the given vector \( x \) onto the null space of \( A \) and name this vector \( \vec{x}_{\text{Nul}} \).
3. Find a basis for the row space of \( A \), denoted as Row(A).
4. Project the vector \( x \) onto the row space of \( A \) and name this vector \( \vec{x}_{\text{Row}} \).
5. Demonstrate that the vectors \( \vec{x}_{\text{Nul}} \) and \( \vec{x}_{\text{Row}} \) are orthogonal and discuss whether this orthogonality would hold for any matrix \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F644afe59-5ebd-4464-95f6-cbe6de6ea324%2Fc2d943fe-4852-487b-8f03-00487f1f888c%2Fs5hx3ha_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Problem Set
Let
\[
x = \begin{bmatrix}
-1 \\
0 \\
2 \\
-2 \\
1
\end{bmatrix}
\quad \text{and} \quad
A = \begin{bmatrix}
3 & 0 & 3 & 0 & 0 \\
0 & -2 & -2 & 0 & 0 \\
0 & -4 & -4 & 0 & 1 \\
3 & -4 & -1 & 0 & 1 \\
1 & 0 & 1 & 0 & 0
\end{bmatrix}
\]
(a) **Find a basis for Nul(A).**
(b) **Find the projection of** \( \vec{x} \) **onto Nul(A). Call this vector** \( \vec{x}_{\text{Nul}} \).
(c) **Find a basis for Row(A).**
(d) **Find the projection of** \( \vec{x} \) **onto Row(A). Call this vector** \( \vec{x}_{\text{Row}} \).
(e) **Show that** \( \vec{x}_{\text{Nul}} \) **and** \( \vec{x}_{\text{Row}} \) **are orthogonal. Will this be the case for all matrices** \( A \) **and all vectors** \( \vec{x} \)? **Explain.**
---
### Explanation of the Diagram
The image provided is a linear algebra problem involving a vector \( x \) and a matrix \( A \). The tasks are to:
1. Determine a basis for the null space of \( A \), denoted as Nul(A).
2. Project the given vector \( x \) onto the null space of \( A \) and name this vector \( \vec{x}_{\text{Nul}} \).
3. Find a basis for the row space of \( A \), denoted as Row(A).
4. Project the vector \( x \) onto the row space of \( A \) and name this vector \( \vec{x}_{\text{Row}} \).
5. Demonstrate that the vectors \( \vec{x}_{\text{Nul}} \) and \( \vec{x}_{\text{Row}} \) are orthogonal and discuss whether this orthogonality would hold for any matrix \(
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Step 1: Related to guidelines
VIEWStep 2: Given information,
VIEWStep 3: Now we reduce the matrix to reduced row echelon form,
VIEWStep 4: we solve to get reduced row echelon form,
VIEWStep 5: we get the rank of a matrix and solve the matrix equation,
VIEWStep 6: finally we get the basis for Nul(A) ,
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