### Vector and Matrix

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 x1A + x2B + x3C = X,

 x1[ 1 2 ] + x2[ 1 0 ] + x3[ 0 1 ] = [ 3 4 ] 0 1 2 1 -1 0 2 3

we have the following system of equations.

 x1 + x2 = 3 2x1 + x3 = 4 2x2 - x3 = 2 x1 + x2 = 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 x1 = 2, x2 = 1, x3 = 0. Therefore X = 2A + 1B + 0C. Moreover, the expression is not unique.

For a general 2 by 2 matrix X = (bij) (i.e., bij is the entry of X at the i-th row and j-th column), the corresponding system is

 x1 + x2 = b11 2x1 + x3 = b12 2x2 - x3 = b21 x1 + x2 = b22

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.