t Let A be a 12x9 matrix whose transpose A has bullity 4. What is the maximum number of linearly independent vectors in the column space of A?
t Let A be a 12x9 matrix whose transpose A has bullity 4. What is the maximum number of linearly independent vectors in the column space of A?
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 3CEXP
Related questions
Question
![**Problem Statement:**
Let \( A \) be a \( 12 \times 9 \) matrix whose transpose \( A^T \) has nullity 4. What is the maximum number of linearly independent vectors in the column space of \( A \)?
**Explanation:**
To solve this problem, we need to use the fundamental theorem of linear algebra, which relates the rank and nullity of a matrix.
1. We are given that \( \text{nullity}(A^T) = 4 \).
2. The nullity of a matrix \( A \) is the dimension of the null space of \( A \), which is the number of linearly independent solutions to the equation \( A\mathbf{x} = 0 \).
For the transpose of the matrix:
\[ \text{nullity}(A^T) + \text{rank}(A^T) = \text{number of columns in } A^T \]
Since \( A \) is a \( 12 \times 9 \) matrix:
- \( A^T \) is a \( 9 \times 12 \) matrix.
- Therefore, the number of columns in \( A^T \) is 12.
So,
\[ 4 + \text{rank}(A^T) = 12 \]
Thus,
\[ \text{rank}(A^T) = 8 \]
The rank of \( A^T \) is the same as the rank of \( A \), which means:
\[ \text{rank}(A) = 8 \]
The rank of a matrix is the dimension of the column space of the matrix, which is the maximum number of linearly independent vectors in the column space.
**Answer:**
The maximum number of linearly independent vectors in the column space of \( A \) is 8.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc37c896-f9bf-421c-983e-bcaad18451c5%2Fa134dde1-9702-41ea-ab39-08a5eeae6337%2Fu1tjlj_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( A \) be a \( 12 \times 9 \) matrix whose transpose \( A^T \) has nullity 4. What is the maximum number of linearly independent vectors in the column space of \( A \)?
**Explanation:**
To solve this problem, we need to use the fundamental theorem of linear algebra, which relates the rank and nullity of a matrix.
1. We are given that \( \text{nullity}(A^T) = 4 \).
2. The nullity of a matrix \( A \) is the dimension of the null space of \( A \), which is the number of linearly independent solutions to the equation \( A\mathbf{x} = 0 \).
For the transpose of the matrix:
\[ \text{nullity}(A^T) + \text{rank}(A^T) = \text{number of columns in } A^T \]
Since \( A \) is a \( 12 \times 9 \) matrix:
- \( A^T \) is a \( 9 \times 12 \) matrix.
- Therefore, the number of columns in \( A^T \) is 12.
So,
\[ 4 + \text{rank}(A^T) = 12 \]
Thus,
\[ \text{rank}(A^T) = 8 \]
The rank of \( A^T \) is the same as the rank of \( A \), which means:
\[ \text{rank}(A) = 8 \]
The rank of a matrix is the dimension of the column space of the matrix, which is the maximum number of linearly independent vectors in the column space.
**Answer:**
The maximum number of linearly independent vectors in the column space of \( A \) is 8.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 21 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
![College Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305652231/9781305652231_smallCoverImage.gif)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
![College Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305652231/9781305652231_smallCoverImage.gif)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![College Algebra](https://www.bartleby.com/isbn_cover_images/9781938168383/9781938168383_smallCoverImage.gif)
![Elementary Linear Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305658004/9781305658004_smallCoverImage.gif)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
![Intermediate Algebra](https://www.bartleby.com/isbn_cover_images/9780998625720/9780998625720_smallCoverImage.gif)