Which matrix is a linear combination of 1 0 1 0 and 0 0 0 1

Which matrix is a linear combination of [1 0 1 0] and [0 0 0 1]? [1 0 2 3] [0 0 1 0] [2 0 2 3] [1 0 0 1] None of the above

Solution

2 4 1 1 0 3 5 v D 2 4 0 1 1 3 5 w D 2 4 0 0 1 3 5 . Their linear combinations in three-dimensional space are cu C dv C ew: Combinations c 2 4 1 1 0 3 5 C d 2 4 0 1 1 3 5 C e 2 4 0 0 1 3 5 D 2 4 c d c e d 3 5 . (1) Now something important: Rewrite that combination using a matrix. The vectors u; v; w go into the columns of the matrix A. That matrix “multiplies” a vector: Same combination is now A times x 2 4 1 0 0 1 1 0 0 1 1 3 5 2 4 c d e 3 5 D 2 4 c d c e d 3 5 . (2) The numbers c; d; e are the components of a vector x. The matrix A times the vector x is the same as the combination cu C dv C ew of the three columns: Matrix times vector Ax D 2 4 u v w 3 5 2 4 c d e 3 5 D cu C dv C ew . (3) This is more than a definition of Ax, because the rewriting brings a crucial change in viewpoint. At first, the numbers c; d; e were multiplying the vectors. Now the matrix is multiplying those numbers. The matrix A acts on the vector x. The result Ax is a combination b of the columns of A.