Let A and B be matrices of the form: |b1 b12 b3 a 12 b21 b2 b23 A= B= b31 b32 b33 a21 a22 d13 b, b 42 43 Consider M(A) = BA (matrix multiplication). Is M(A) a linear transformation?
Let A and B be matrices of the form: |b1 b12 b3 a 12 b21 b2 b23 A= B= b31 b32 b33 a21 a22 d13 b, b 42 43 Consider M(A) = BA (matrix multiplication). Is M(A) a linear transformation?
Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Matrix Multiplication and Linear Transformations**
Let A and B be matrices of the form:
Matrix A:
\[
A = \begin{pmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
a_{13} & a_{32}
\end{pmatrix}
\]
Matrix B:
\[
B = \begin{pmatrix}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} & b_{32} & b_{33} \\
b_{41} & b_{42} & b_{43}
\end{pmatrix}
\]
Consider \( M(A) = BA \) (matrix multiplication).
Is \( M(A) \) a linear transformation?
**Discussion:**
The problem asks us to determine if the matrix multiplication \( M(A) = BA \) results in a linear transformation. Generally, a linear transformation satisfies two properties:
1. Additivity: \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \)
2. Homogeneity: \( T(c\mathbf{u}) = cT(\mathbf{u}) \)
Matrix multiplication itself is a linear operation. To check if \( M(A) \) is a linear transformation, we would verify that it adheres to these properties using the structure and elements of matrices A and B.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F486b24c8-8b02-4a6d-85d6-6aef3a827ba3%2Fb6717f69-cd36-4856-8d79-a63a3d438d62%2Fg6i6fnz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Matrix Multiplication and Linear Transformations**
Let A and B be matrices of the form:
Matrix A:
\[
A = \begin{pmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
a_{13} & a_{32}
\end{pmatrix}
\]
Matrix B:
\[
B = \begin{pmatrix}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} & b_{32} & b_{33} \\
b_{41} & b_{42} & b_{43}
\end{pmatrix}
\]
Consider \( M(A) = BA \) (matrix multiplication).
Is \( M(A) \) a linear transformation?
**Discussion:**
The problem asks us to determine if the matrix multiplication \( M(A) = BA \) results in a linear transformation. Generally, a linear transformation satisfies two properties:
1. Additivity: \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \)
2. Homogeneity: \( T(c\mathbf{u}) = cT(\mathbf{u}) \)
Matrix multiplication itself is a linear operation. To check if \( M(A) \) is a linear transformation, we would verify that it adheres to these properties using the structure and elements of matrices A and B.
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