Exercise Express the matrix
X = [ | 3 | 4 | ] |
2 | 3 |
as linear combinations of the matrices.
A = [ | 1 | 2 | ], B = [ | 1 | 0 | ], C = [ | 0 | 1 | ] |
0 | 1 | 2 | 1 | -1 | 0 |
Is the expression (i.e., the coefficients) unique? Is it true that any 2 by 2 matrix is a linear combination of A, B and C?
Answer By comparing the four entries on the both side of the equation x_{1}A + x_{2}B + x_{3}C = X,
x_{1}[ | 1 | 2 | ] + x_{2}[ | 1 | 0 | ] + x_{3}[ | 0 | 1 | ] = [ | 3 | 4 | ] |
0 | 1 | 2 | 1 | -1 | 0 | 2 | 3 |
we have the following system of equations.
x_{1} | + x_{2} | = | 3 | |
2x_{1} | + x_{3} | = | 4 | |
2x_{2} | - x_{3} | = | 2 | |
x_{1} | + x_{2} | = | 3 |
From the row echelon form of the augmented matrix,
[ | 1 | 1 | 0 | 3 | ] |
0 | 2 | -1 | 2 | ||
0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 |
we see that the system has non-unique solutions. One solution is x_{1} = 2, x_{2} = 1, x_{3} = 0. Therefore X = 2A + 1B + 0C. Moreover, the expression is not unique.
For a general 2 by 2 matrix X = (b_{ij}) (i.e., b_{ij} is the entry of X at the i-th row and j-th column), the corresponding system is
x_{1} | + x_{2} | = | b_{11} | |
2x_{1} | + x_{3} | = | b_{12} | |
2x_{2} | - x_{3} | = | b_{21} | |
x_{1} | + x_{2} | = | b_{22} |
Then from the row echelon form above (or by the fact that there are more equations than variables), we see that solutions do not always exist. In other words, not any X can be expressed as a linear combination of A, B and C.
Remark In an earlier exercise, we have seen how the problem of expressing a vector as a linear combination of some other vectors is equivalent to solving a system of linear equations. This exercise shows that we have similar relation for matrices.