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1. Suppose A is an 8 by 11 matrix with rank 5. (a) Give the dimensions of the 4 basic subspaces we studied, and explain how you found them. (b) How many solutions might there be to Ax = b? How many solutions to A'y = c. Explain your answers.

1. Suppose A is an 8 by 11 matrix with rank 5. (a) Give the dimensions of the 4 basic subspaces we studied, and explain how you found them. (b) How many solutions might there be to Ax = b? How many solutions to A'y = c. Explain your answers.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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For part a, fine row space, column space, null space, left null space

### Linear Algebra Exercise **Problem Statement:** 1. Suppose \( A \) is an 8 by 11 matrix with rank 5. **Questions:** (a) Give the dimensions of the 4 basic subspaces we studied, and explain how you found them. (b) How many solutions might there be to \( Ax = b \)? How many solutions to \( A^Ty = c \)? Explain your answers. **Detailed Explanation:** This question involves understanding the fundamental subspaces of a matrix and the dimensions of these subspaces. The four basic subspaces related to the matrix \( A \) are: 1. **Column Space (C(A)):** - The dimension of the column space of \( A \) is equal to the rank of \( A \). - For this matrix \( A \) (8 by 11) with a rank of 5, the dimension of the column space is 5. 2. **Null Space (N(A)):** - The dimension of the null space of \( A \) is given by \( \text{number of columns} - \text{rank} \). - For this matrix, the number of columns is 11, so the dimension of the null space is \( 11 - 5 = 6 \). 3. **Row Space (C(A^T)):** - The dimension of the row space of \( A \) is also equal to the rank of \( A \). - Hence, the dimension of the row space is 5. 4. **Left Null Space (N(A^T)):** - The dimension of the left null space of \( A \) is given by \( \text{number of rows} - \text{rank} \). - For this matrix, the number of rows is 8, so the dimension of the left null space is \( 8 - 5 = 3 \). **Solutions for \( Ax = b \) and \( A^Ty = c \):** - **Solutions to \( Ax = b \):** - There are three possible cases: - **No solutions** if \( b \) is not in the column space of \( A \). - **One solution** if \( b \) is in the column space and the null space is trivial. - **Infinitely many solutions** if \( b \) is in the column space
Transcribed Image Text:### Linear Algebra Exercise **Problem Statement:** 1. Suppose \( A \) is an 8 by 11 matrix with rank 5. **Questions:** (a) Give the dimensions of the 4 basic subspaces we studied, and explain how you found them. (b) How many solutions might there be to \( Ax = b \)? How many solutions to \( A^Ty = c \)? Explain your answers. **Detailed Explanation:** This question involves understanding the fundamental subspaces of a matrix and the dimensions of these subspaces. The four basic subspaces related to the matrix \( A \) are: 1. **Column Space (C(A)):** - The dimension of the column space of \( A \) is equal to the rank of \( A \). - For this matrix \( A \) (8 by 11) with a rank of 5, the dimension of the column space is 5. 2. **Null Space (N(A)):** - The dimension of the null space of \( A \) is given by \( \text{number of columns} - \text{rank} \). - For this matrix, the number of columns is 11, so the dimension of the null space is \( 11 - 5 = 6 \). 3. **Row Space (C(A^T)):** - The dimension of the row space of \( A \) is also equal to the rank of \( A \). - Hence, the dimension of the row space is 5. 4. **Left Null Space (N(A^T)):** - The dimension of the left null space of \( A \) is given by \( \text{number of rows} - \text{rank} \). - For this matrix, the number of rows is 8, so the dimension of the left null space is \( 8 - 5 = 3 \). **Solutions for \( Ax = b \) and \( A^Ty = c \):** - **Solutions to \( Ax = b \):** - There are three possible cases: - **No solutions** if \( b \) is not in the column space of \( A \). - **One solution** if \( b \) is in the column space and the null space is trivial. - **Infinitely many solutions** if \( b \) is in the column space
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