7. Find the values of p and q so that Nul A is a subspace of RP and Col A is a subspace of R'. 4 A = 1 -2 6. 1 1
7. Find the values of p and q so that Nul A is a subspace of RP and Col A is a subspace of R'. 4 A = 1 -2 6. 1 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
![**Problem 7: Determining Subspace Dimensions**
Find the values of \( p \) and \( q \) so that \(\text{Nul } A\) is a subspace of \(\mathbb{R}^p\) and \(\text{Col } A\) is a subspace of \(\mathbb{R}^q\).
The matrix \( A \) is given by:
\[
A = \begin{bmatrix}
4 & 5 & -2 & 6 & 0 \\
1 & 1 & 0 & 1 & 0
\end{bmatrix}
\]
**Explanation:**
This problem requires you to determine the appropriate dimensions for the null space and column space of the matrix \( A \), based on its structure. The row count of \( A \) determines the dimension related to the column space, while the column count informs the dimension for the null space.
- **Null Space (\(\text{Nul } A\))**: The null space of a matrix \( A \) consists of all vectors \( x \) such that \( Ax = 0 \). The dimension of the null space is calculated as the number of columns minus the rank of the matrix.
- **Column Space (\(\text{Col } A\))**: The column space consists of all possible linear combinations of the column vectors of \( A \). The dimension of the column space is equal to the rank of the matrix.
In this case, identify the rank of \( A \) to find the dimensions of both spaces, and subsequently, the values of \( p \) and \( q \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2191ff5f-0867-4682-9559-0b1c4e7ecbac%2F447306af-0260-460b-ab5f-064e70345bfc%2Fbi2qrow_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 7: Determining Subspace Dimensions**
Find the values of \( p \) and \( q \) so that \(\text{Nul } A\) is a subspace of \(\mathbb{R}^p\) and \(\text{Col } A\) is a subspace of \(\mathbb{R}^q\).
The matrix \( A \) is given by:
\[
A = \begin{bmatrix}
4 & 5 & -2 & 6 & 0 \\
1 & 1 & 0 & 1 & 0
\end{bmatrix}
\]
**Explanation:**
This problem requires you to determine the appropriate dimensions for the null space and column space of the matrix \( A \), based on its structure. The row count of \( A \) determines the dimension related to the column space, while the column count informs the dimension for the null space.
- **Null Space (\(\text{Nul } A\))**: The null space of a matrix \( A \) consists of all vectors \( x \) such that \( Ax = 0 \). The dimension of the null space is calculated as the number of columns minus the rank of the matrix.
- **Column Space (\(\text{Col } A\))**: The column space consists of all possible linear combinations of the column vectors of \( A \). The dimension of the column space is equal to the rank of the matrix.
In this case, identify the rank of \( A \) to find the dimensions of both spaces, and subsequently, the values of \( p \) and \( q \).
Expert Solution
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 2 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
