Question

Determine a basis for the subspace of M2(R) spanned by the following. (Enter your answer as a list of matrices. Enter each matrix as a comma-separated list of its components in the form [[a11, a12l, [a21, a221.) 5-10 1 1 1 L-51 -1 2

Answer 1

To find a basis for the subspace spanned by the given matrices, we need to put them into row echelon form and then choose the non-zero rows as our basis. We start by forming the augmented matrix and reducing it:

[ 5 -1 1 | 0 ]
[-10 1 -1 | 0 ]
[ 1 -1 2 | 0 ]

R2 -> R2 + 2R1
R3 -> R3 - (1/5)R1

[ 5 -1 1 | 0 ]
[ 0 1 -1 | 0 ]
[ 0 -4/5 9/5| 0 ]

R3 -> R3 + (4/5)R2

[ 5 -1 1 | 0 ]
[ 0 1 -1 | 0 ]
[ 0 0 5/3| 0 ]

Now we see that the third row corresponds to the equation (5/3)z = 0, which implies that z = 0. Therefore, the subspace is spanned by the two matrices:

[ 5 -1 1 ]
[ 0 1 -1 ]

We can check that these matrices are linearly independent, which means that they form a basis for the subspace. Therefore, the basis is:

[[5, -1, 1], [0, 1, -1]]