Week 12: Part 1: This exercise will ask you to explore the equality Null(A") = (Col(A))-. Pick a vector u in Rª and a vector v such that u and v are orthogonal; then find a vector w which rence and Technology Onieasied interview 7 is orthogonal to both u and v. Finally, find a vector z which is orthogonal to u, v, w. For the last two steps, you should realize that you are solving a system of linear equations, and if write them down that the system is represented by AT where the columns of A are the vectors you are trying to be orthogonal to. Now, verify with computations that the set {u, v, w, z} is linearly independent. If any of the vectors u, v, w, z are scalars of the standard basis vectors e1, e2, e3, e4 then start over. Set the matrix P = [u v w z] and compute without calculations the vectors P-lu, P-lv, P-lw, and P-lz. you
Week 12: Part 1: This exercise will ask you to explore the equality Null(A") = (Col(A))-. Pick a vector u in Rª and a vector v such that u and v are orthogonal; then find a vector w which rence and Technology Onieasied interview 7 is orthogonal to both u and v. Finally, find a vector z which is orthogonal to u, v, w. For the last two steps, you should realize that you are solving a system of linear equations, and if write them down that the system is represented by AT where the columns of A are the vectors you are trying to be orthogonal to. Now, verify with computations that the set {u, v, w, z} is linearly independent. If any of the vectors u, v, w, z are scalars of the standard basis vectors e1, e2, e3, e4 then start over. Set the matrix P = [u v w z] and compute without calculations the vectors P-lu, P-lv, P-lw, and P-lz. you
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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![**Week 12: Part 1:**
*This exercise will ask you to explore the equality* \( \text{Null}(A^T) = (\text{Col}(A))^\perp \).
1. Pick a vector \( \mathbf{u} \) in \( \mathbb{R}^4 \) and a vector \( \mathbf{v} \) such that \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal; then find a vector \( \mathbf{w} \) which is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \).
2. Finally, find a vector \( \mathbf{z} \) which is orthogonal to \( \mathbf{u}, \mathbf{v}, \mathbf{w} \).
For the last two steps, you should realize that you are solving a system of linear equations, and if you write them down, that the system is represented by \( A^T \) where the columns of \( A \) are the vectors you are trying to be orthogonal to.
- Now, verify with computations that the set \{ \( \mathbf{u}, \mathbf{v}, \mathbf{w}, \mathbf{z} \} \) is linearly independent.
- If any of the vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w}, \mathbf{z} \) are scalars of the standard basis vectors \( \mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3}, \mathbf{e_4} \), then start over.
3. Set the matrix \( P = [\mathbf{u} \ \mathbf{v} \ \mathbf{w} \ \mathbf{z}] \) and compute *without calculations* the vectors \( P^{-1}\mathbf{u}, P^{-1}\mathbf{v}, P^{-1}\mathbf{w}, \) and \( P^{-1}\mathbf{z} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ce5f382-c21d-48ef-8c6b-42079a16ed57%2F3f5eaa43-a903-4b01-8c7c-e18fdd8eccab%2Fexk0ho_processed.png&w=3840&q=75)
Transcribed Image Text:**Week 12: Part 1:**
*This exercise will ask you to explore the equality* \( \text{Null}(A^T) = (\text{Col}(A))^\perp \).
1. Pick a vector \( \mathbf{u} \) in \( \mathbb{R}^4 \) and a vector \( \mathbf{v} \) such that \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal; then find a vector \( \mathbf{w} \) which is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \).
2. Finally, find a vector \( \mathbf{z} \) which is orthogonal to \( \mathbf{u}, \mathbf{v}, \mathbf{w} \).
For the last two steps, you should realize that you are solving a system of linear equations, and if you write them down, that the system is represented by \( A^T \) where the columns of \( A \) are the vectors you are trying to be orthogonal to.
- Now, verify with computations that the set \{ \( \mathbf{u}, \mathbf{v}, \mathbf{w}, \mathbf{z} \} \) is linearly independent.
- If any of the vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w}, \mathbf{z} \) are scalars of the standard basis vectors \( \mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3}, \mathbf{e_4} \), then start over.
3. Set the matrix \( P = [\mathbf{u} \ \mathbf{v} \ \mathbf{w} \ \mathbf{z}] \) and compute *without calculations* the vectors \( P^{-1}\mathbf{u}, P^{-1}\mathbf{v}, P^{-1}\mathbf{w}, \) and \( P^{-1}\mathbf{z} \).
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