Can Any Matrix Be Expressed as a Linear Combination of Other Matrices?

[hkus10]

1) Is the matrix [upper row 3 0 and lower row 0 2] a linear combination of the matrices [upper row 1 0 and lower row 0 1] and [upper row 1 0 and lower row 0 0]? Justify your answer.

Is it I just have to add the two matrices to see if they are equal the matrix, [upper row 3 0 and lower row 0 2]?

2) Show that the linear system obtained by adding a multiple of an equation in (2) to another equation is equivalent to (2).

How to show that?

Thanks!

 

[hotvette]

Is it I just have to add the two matrices to see if they are equal the matrix, [upper row 3 0 and lower row 0 2]?

Not exactly. If A,B, C are matrices, A is a linear combination of B & C if A = a*B + b*C where a and b are scalars. Try it. You'll get your answer pretty quickly.