Give bases for row(A), col(A), and null(A). 1 1 -4 0 2 1 1 -1 -5 row (A) col(A) null(A) A =
Give bases for row(A), col(A), and null(A). 1 1 -4 0 2 1 1 -1 -5 row (A) col(A) null(A) A =
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
![### Bases for Row Space, Column Space, and Null Space
#### Matrix \( A \)
The given matrix \( A \) is:
\[
A = \begin{bmatrix}
1 & 1 & -4 \\
0 & 2 & 1 \\
1 & -1 & -5
\end{bmatrix}
\]
#### Row Space \(\text{row}(A)\)
The row space of a matrix consists of all possible linear combinations of its row vectors. The basis for the row space of matrix \( A \) can be obtained by selecting a set of linearly independent row vectors.
Here, the row space is represented with vectors showing placeholders for calculating the basis.
#### Column Space \(\text{col}(A)\)
The column space of a matrix is defined as the span of the column vectors of the matrix. It is represented by vectors with placeholders for determining which columns form a basis for the column space.
#### Null Space \(\text{null}(A)\)
The null space of a matrix consists of all the vectors \( x \) such that \( Ax = 0 \). The basis for the null space can be obtained by solving the homogeneous equation \( Ax = 0 \).
Vectors with placeholders indicate steps to find the basis for the null space.
#### Visual Representation
- The image uses green arrows for directional guidance on how to perform row operations.
- Rectangular brackets illustrate how bases would be represented once calculated.
- The matrices indicate the transformation or selection process for each respective space (row, column, null).
This educational content helps in identifying and understanding the basis of the different vector spaces associated with a given matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6948d5d-782c-4f32-af22-abf47b4c3397%2F9ba61729-f0a7-491e-a47a-6404ceaa8f7b%2Fz22i4ar_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Bases for Row Space, Column Space, and Null Space
#### Matrix \( A \)
The given matrix \( A \) is:
\[
A = \begin{bmatrix}
1 & 1 & -4 \\
0 & 2 & 1 \\
1 & -1 & -5
\end{bmatrix}
\]
#### Row Space \(\text{row}(A)\)
The row space of a matrix consists of all possible linear combinations of its row vectors. The basis for the row space of matrix \( A \) can be obtained by selecting a set of linearly independent row vectors.
Here, the row space is represented with vectors showing placeholders for calculating the basis.
#### Column Space \(\text{col}(A)\)
The column space of a matrix is defined as the span of the column vectors of the matrix. It is represented by vectors with placeholders for determining which columns form a basis for the column space.
#### Null Space \(\text{null}(A)\)
The null space of a matrix consists of all the vectors \( x \) such that \( Ax = 0 \). The basis for the null space can be obtained by solving the homogeneous equation \( Ax = 0 \).
Vectors with placeholders indicate steps to find the basis for the null space.
#### Visual Representation
- The image uses green arrows for directional guidance on how to perform row operations.
- Rectangular brackets illustrate how bases would be represented once calculated.
- The matrices indicate the transformation or selection process for each respective space (row, column, null).
This educational content helps in identifying and understanding the basis of the different vector spaces associated with a given matrix.
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 3 steps with 3 images
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,
