91Ó°ÊÓ

Understanding linear combinations is key to grasping the concept of linear independence. A linear combination involves combining multiple elements, such as vectors or matrices, by multiplying each one by a constant and then summing the results. In our exercise, the matrices were combined as follows: \( c_1A_1 + c_2A_2 + c_3A_3 = 0 \).
A linear combination is often used to determine if a set of elements is independent by examining if they can express each other. When matrices, for instance, can be written as a linear combination of others, it indicates dependence. In our example:

• Matrix 1: \( A_1 = \begin{bmatrix} 1 & 1 \ 0 & -1 \end{bmatrix} \)
• Matrix 2: \( A_2 = \begin{bmatrix} 1 & -1 \ 1 & 0 \end{bmatrix} \)
• Matrix 3: \( A_3 = \begin{bmatrix} 1 & 0 \ 3 & 2 \end{bmatrix} \)

The test for independence involves finding if all coefficients \( c_1, c_2, \) and \( c_3 \) are zero in a non-trivial solution. A non-zero solution indicates dependency, showing how one or more elements are redundant and do not add new information to the system.

Matrix equations help us express complex relationships between matrices and can aid in testing linear independence. A matrix equation, in general, is an expression where matrices are equal, often used to solve unknowns. In the exercise, matrices \( A_1, A_2, \) and \( A_3 \) were used to form a matrix equation.
Substituting into our equation:\[c_1\begin{bmatrix} 1 & 1 \ 0 & -1 \end{bmatrix} + c_2\begin{bmatrix} 1 & -1 \ 1 & 0 \end{bmatrix} + c_3\begin{bmatrix} 1 & 0 \ 3 & 2 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}\]This equation describes how combinations of the matrices add to zero when balanced correctly through their coefficients. Such equations are critical in understanding the nature of matrices and their interdependencies. Being able to set up these equations correctly is vital to address problems in linear algebra, offering a foundation to solve for unknown variables and verifying conditions such as linear dependence.

The solution of systems of equations involves finding values that satisfy all given equations simultaneously. In testing for linear independence, this often means evaluating if all coefficients must be zero. From the previous step of combining equations, we derived the system:
1. \( c_1 + c_2 + c_3 = 0 \)
2. \( c_1 - c_2 = 0 \)
3. \( c_2 + 3c_3 = 0 \)
4. \( -c_1 + 2c_3 = 0 \)
Simplifying these equations is essential in understanding the dependence or independence of the system. In our solution:

• Using equation 2, we solved for \( c_1 \) as \( c_1 = c_2 \).
• Equation 3 gave us \( c_2 = -3c_3 \).
• Substituting all knowns back showed \( c_3 = 0 \), resulting in \( c_1 = c_2 = 0 \).

This solution demonstrates that the initial test for dependency was incorrect; in fact, the matrices are linearly independent. Understanding why each variable resolves to zero helps affirm that no matrix in the set can be expressed as a combination of the others.