Determine whether or not the following sets s of 2×2 matrices are linearly independent.
The Correct Answer and Explanation is:
To determine if a set of 2×2 matrices is linearly independent, we need to check whether a linear combination of the matrices can result in the zero matrix only if all the coefficients in the combination are zero. In other words, if the set of matrices is linearly independent, the only solution to the equation
[
c_1 A_1 + c_2 A_2 + \dots + c_n A_n = 0
]
should be (c_1 = c_2 = \dots = c_n = 0), where (A_1, A_2, \dots, A_n) are the matrices in the set, and (c_1, c_2, \dots, c_n) are scalar coefficients.
Steps to check for linear independence of matrices:
- Set up the equation:
We assume that we have a linear combination of matrices in the set equal to the zero matrix. That is: [
c_1 A_1 + c_2 A_2 + \dots + c_n A_n = 0
] - Express the matrices as vectors:
Each 2×2 matrix can be expressed as a vector of its four entries. For instance, a matrix (A = \begin{pmatrix} a & b \ c & d \end{pmatrix}) is treated as the vector ((a, b, c, d)). This step converts the matrix equation into a system of linear equations in terms of the scalars (c_1, c_2, \dots, c_n). - Solve the system:
Solve the system of linear equations formed by equating the linear combination of matrices to the zero matrix. This can be done using Gaussian elimination or matrix methods to check if the only solution is (c_1 = c_2 = \dots = c_n = 0). - Conclusion:
If the only solution to the system is the trivial solution (all coefficients are zero), then the set of matrices is linearly independent. If there is a non-trivial solution, the set is linearly dependent.
Example:
Consider the set of matrices:
[
A_1 = \begin{pmatrix} 1 & 0 \ 0 & 0 \end{pmatrix}, \quad A_2 = \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}, \quad A_3 = \begin{pmatrix} 0 & 0 \ 1 & 0 \end{pmatrix}, \quad A_4 = \begin{pmatrix} 0 & 0 \ 0 & 1 \end{pmatrix}
]
We write the equation ( c_1 A_1 + c_2 A_2 + c_3 A_3 + c_4 A_4 = 0 ):
[
c_1 \begin{pmatrix} 1 & 0 \ 0 & 0 \end{pmatrix} + c_2 \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix} + c_3 \begin{pmatrix} 0 & 0 \ 1 & 0 \end{pmatrix} + c_4 \begin{pmatrix} 0 & 0 \ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}
]
This expands to:
[
\begin{pmatrix} c_1 & c_2 \ c_3 & c_4 \end{pmatrix} = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}
]
This results in the system of equations:
[
c_1 = 0, \quad c_2 = 0, \quad c_3 = 0, \quad c_4 = 0
]
Since the only solution is ( c_1 = c_2 = c_3 = c_4 = 0 ), the set of matrices is linearly independent.
Thus, if the system only has the trivial solution, the set of matrices is linearly independent. If there exists a non-trivial solution, the set is linearly dependent.