5. Consider the 4 x 5 matrix A = [u₁|u2|u3|u4| u5], where the columns are U₁= 2 , U2 = 5 Uz = -1 -2 3 U4 = U5 = a) Find a set of vectors in {u1, U2, U3, U4, us} which is a basis of the column space of A. b) Find the rank of A.
5. Consider the 4 x 5 matrix A = [u₁|u2|u3|u4| u5], where the columns are U₁= 2 , U2 = 5 Uz = -1 -2 3 U4 = U5 = a) Find a set of vectors in {u1, U2, U3, U4, us} which is a basis of the column space of A. b) Find the rank of A.
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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![**Matrix Problem 5: Understanding and Solving a Column Space Basis**
Consider the 4×5 matrix \( A = [u_1 \, | \, u_2 \, | \, u_3 \, | \, u_4 \, | \, u_5] \), where the columns are defined as follows:
\[
u_1 = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}, \quad
u_2 = \begin{bmatrix} 5 \\ 1 \\ 0 \\ 2 \end{bmatrix}, \quad
u_3 = \begin{bmatrix} 2 \\ -1 \\ -2 \\ 3 \end{bmatrix}, \quad
u_4 = \begin{bmatrix} 1 \\ 0 \\ -1 \\ 1 \end{bmatrix}, \quad
u_5 = \begin{bmatrix} 1 \\ 2 \\ 2 \\ 5 \end{bmatrix}
\]
**Tasks:**
a) Identify a set of vectors in \(\{u_1, u_2, u_3, u_4, u_5\}\) that forms a basis for the column space of \( A \).
b) Determine the rank of \( A \).
**Explanation of Tasks:**
- **Column Space Basis:** To find a basis for the column space, you need a set of linearly independent vectors from the columns of matrix \( A \) that span the entire column space.
- **Rank:** The rank of \( A \) is determined by the number of linearly independent columns and corresponds to the dimension of the column space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2F63e0c64a-6d53-403a-906f-9aba4fd0e5f8%2Fsow1wnj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Matrix Problem 5: Understanding and Solving a Column Space Basis**
Consider the 4×5 matrix \( A = [u_1 \, | \, u_2 \, | \, u_3 \, | \, u_4 \, | \, u_5] \), where the columns are defined as follows:
\[
u_1 = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}, \quad
u_2 = \begin{bmatrix} 5 \\ 1 \\ 0 \\ 2 \end{bmatrix}, \quad
u_3 = \begin{bmatrix} 2 \\ -1 \\ -2 \\ 3 \end{bmatrix}, \quad
u_4 = \begin{bmatrix} 1 \\ 0 \\ -1 \\ 1 \end{bmatrix}, \quad
u_5 = \begin{bmatrix} 1 \\ 2 \\ 2 \\ 5 \end{bmatrix}
\]
**Tasks:**
a) Identify a set of vectors in \(\{u_1, u_2, u_3, u_4, u_5\}\) that forms a basis for the column space of \( A \).
b) Determine the rank of \( A \).
**Explanation of Tasks:**
- **Column Space Basis:** To find a basis for the column space, you need a set of linearly independent vectors from the columns of matrix \( A \) that span the entire column space.
- **Rank:** The rank of \( A \) is determined by the number of linearly independent columns and corresponds to the dimension of the column space.
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