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5.12. Give a basis find its dimension. elements). for the vector space of all 3 x 3 matrices of trace 0 and (Recall that the trace of a matrix is the sum of its diagonal

5.12. Give a basis find its dimension. elements). for the vector space of all 3 x 3 matrices of trace 0 and (Recall that the trace of a matrix is the sum of its diagonal

Linear Algebra: A Modern Introduction
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
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### 5.12 Dimension is 8 To avoid writing out all the eight matrices in the basis explicitly, let us define \( E^{(ij)} \) to be the \( 3 \times 3 \) matrix whose \( ij \)-th entry is 1 and all its other entries are 0. For example, \( E^{(23)} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \). Then a basis is given by: \[ \{ E^{(12)}, E^{(13)}, E^{(21)}, E^{(23)}, E^{(31)}, E^{(32)}, E^{(11)} - E^{(33)}, E^{(22)} - E^{(33)} \}. \]
Transcribed Image Text:### 5.12 Dimension is 8 To avoid writing out all the eight matrices in the basis explicitly, let us define \( E^{(ij)} \) to be the \( 3 \times 3 \) matrix whose \( ij \)-th entry is 1 and all its other entries are 0. For example, \( E^{(23)} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \). Then a basis is given by: \[ \{ E^{(12)}, E^{(13)}, E^{(21)}, E^{(23)}, E^{(31)}, E^{(32)}, E^{(11)} - E^{(33)}, E^{(22)} - E^{(33)} \}. \]
### Problem Statement **5.12.** Give a basis for the vector space of all \( 3 \times 3 \) matrices of trace 0 and find its dimension. (Recall that the trace of a matrix is the sum of its diagonal elements). ### Explanation This problem prompts you to find a basis for the vector space containing all \( 3 \times 3 \) matrices such that the sum of their diagonal elements (known as the trace) is zero. The trace zero condition imposes a constraint that must be satisfied by each matrix in this vector space. Additionally, you are to determine the dimension of this vector space, which corresponds to the number of matrices in your chosen basis. ### Steps to Approach the Problem 1. **Understand the Vector Space**: - Familiarize yourself with the properties of \( 3 \times 3 \) matrices, especially the trace function. - Understand that the trace of a \( 3 \times 3 \) matrix \( A = [a_{ij}] \) is given by \( \text{tr}(A) = a_{11} + a_{22} + a_{33} \). 2. **Define Basis**: - Construct matrices such that each matrix adheres to the trace-zero constraint. - Determine a set of linearly independent matrices that span the entire vector space of \( 3 \times 3 \) trace zero matrices. 3. **Calculate Dimension**: - Count the number of basis matrices found. This count gives the dimension of the vector space. By solving this problem, you will enhance your understanding of linear algebraic concepts like vector spaces, basis, and dimension, and gain practical experience in dealing with matrix operations.
Transcribed Image Text:### Problem Statement **5.12.** Give a basis for the vector space of all \( 3 \times 3 \) matrices of trace 0 and find its dimension. (Recall that the trace of a matrix is the sum of its diagonal elements). ### Explanation This problem prompts you to find a basis for the vector space containing all \( 3 \times 3 \) matrices such that the sum of their diagonal elements (known as the trace) is zero. The trace zero condition imposes a constraint that must be satisfied by each matrix in this vector space. Additionally, you are to determine the dimension of this vector space, which corresponds to the number of matrices in your chosen basis. ### Steps to Approach the Problem 1. **Understand the Vector Space**: - Familiarize yourself with the properties of \( 3 \times 3 \) matrices, especially the trace function. - Understand that the trace of a \( 3 \times 3 \) matrix \( A = [a_{ij}] \) is given by \( \text{tr}(A) = a_{11} + a_{22} + a_{33} \). 2. **Define Basis**: - Construct matrices such that each matrix adheres to the trace-zero constraint. - Determine a set of linearly independent matrices that span the entire vector space of \( 3 \times 3 \) trace zero matrices. 3. **Calculate Dimension**: - Count the number of basis matrices found. This count gives the dimension of the vector space. By solving this problem, you will enhance your understanding of linear algebraic concepts like vector spaces, basis, and dimension, and gain practical experience in dealing with matrix operations.
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