1. (a) Matrix A is 5 by 8 and its rank is 3. What can you say about the four subspaces, their dimensions, the number of solutions to Ax = b, etc.? (b) ) Matrix A is 7 by 9 and its rank is 4. What can you say about the four subspaces, their dimensions, the number of solutions to Ax = b, etc.?
1. (a) Matrix A is 5 by 8 and its rank is 3. What can you say about the four subspaces, their dimensions, the number of solutions to Ax = b, etc.? (b) ) Matrix A is 7 by 9 and its rank is 4. What can you say about the four subspaces, their dimensions, the number of solutions to Ax = b, etc.?
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Linear Algebra Problem Set
#### Question 1
1. (a) Matrix A is 5 by 8 and its rank is 3. What can you say about the four subspaces, their dimensions, the number of solutions to Ax = b, etc.?
(b) Matrix A is 7 by 9 and its rank is 4. What can you say about the four subspaces, their dimensions, the number of solutions to Ax = b, etc.?
---
### Explanation
For each part of this question, you will need to consider the properties and implications of the given matrices.
#### Part (a)
**Matrix A Details:**
- Dimensions: 5 rows by 8 columns
- Rank: 3
When examining the subspaces and related questions:
- **Column Space:** The dimension of the column space (also known as the rank) is 3. This implies that there are 3 linearly independent columns.
- **Null Space:** The nullity, which is the dimension of the null space, can be calculated using the formula: nullity = number of columns - rank. Therefore, the nullity in this case is 8 - 3 = 5.
- **Row Space:** The row space has the same dimension as the rank, which is 3.
- **Left Null Space:** The dimension of the left null space is given by: number of rows - rank. Thus, it is 5 - 3 = 2.
Regarding the solutions to Ax = b:
- **Existence of Solutions:** A general solution requires the right-hand side vector b to lie within the column space of matrix A. Given that the rank is 3 and there are 5 rows, solutions exist if rank(A) = rank([A|b]).
- **Number of Solutions:** If solutions exist, because the null space dimension is 5, there will be infinitely many solutions with a 5-dimensional solution space.
#### Part (b)
**Matrix A Details:**
- Dimensions: 7 rows by 9 columns
- Rank: 4
Evaluating subspaces and related questions:
- **Column Space:** The rank of 4 indicates that the column space has a dimension of 4, implying 4 linearly independent columns.
- **Null Space:** The dimension of the null space is 9 - 4 = 5, as there are 9 columns and the rank is 4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d2fdd51-a813-4b36-89e9-f9581acfc2ee%2Fcad386f9-1543-49d0-b137-da003cafdbbd%2Flru29v_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Problem Set
#### Question 1
1. (a) Matrix A is 5 by 8 and its rank is 3. What can you say about the four subspaces, their dimensions, the number of solutions to Ax = b, etc.?
(b) Matrix A is 7 by 9 and its rank is 4. What can you say about the four subspaces, their dimensions, the number of solutions to Ax = b, etc.?
---
### Explanation
For each part of this question, you will need to consider the properties and implications of the given matrices.
#### Part (a)
**Matrix A Details:**
- Dimensions: 5 rows by 8 columns
- Rank: 3
When examining the subspaces and related questions:
- **Column Space:** The dimension of the column space (also known as the rank) is 3. This implies that there are 3 linearly independent columns.
- **Null Space:** The nullity, which is the dimension of the null space, can be calculated using the formula: nullity = number of columns - rank. Therefore, the nullity in this case is 8 - 3 = 5.
- **Row Space:** The row space has the same dimension as the rank, which is 3.
- **Left Null Space:** The dimension of the left null space is given by: number of rows - rank. Thus, it is 5 - 3 = 2.
Regarding the solutions to Ax = b:
- **Existence of Solutions:** A general solution requires the right-hand side vector b to lie within the column space of matrix A. Given that the rank is 3 and there are 5 rows, solutions exist if rank(A) = rank([A|b]).
- **Number of Solutions:** If solutions exist, because the null space dimension is 5, there will be infinitely many solutions with a 5-dimensional solution space.
#### Part (b)
**Matrix A Details:**
- Dimensions: 7 rows by 9 columns
- Rank: 4
Evaluating subspaces and related questions:
- **Column Space:** The rank of 4 indicates that the column space has a dimension of 4, implying 4 linearly independent columns.
- **Null Space:** The dimension of the null space is 9 - 4 = 5, as there are 9 columns and the rank is 4
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