Is the Set of 3x3 Matrices Annihilating a Given Vector a Subspace?

[KyleS4562]

Homework Statement

Are the 3X3 matrices A such that vector <1,2,3> is in the kernel of A, a subspace of R^(3X3)?

Homework Equations

The Attempt at a Solution

I know that the kernel condition gives a subset V={A|A*<1,2,3>=0} but I am not sure of how to proceed to show it is in fact a subspace. Should I try to see if it spans or use the definition of a subspace? I'm just confused on how to proceed with the next step

 

[aPhilosopher]

To show that it's a subspace, you have to show that if it contains u and v, then it contains au + bv where a and b are scalars, that it contains 0 and that it contains -u for any u that it contains.

 

[Mark44]

To expand slightly on what aPhilosopher said, let U and W be matrices in your subset V. It's sufficient to show that aU + bW is also in the same subset, for any scalars a and b.

Showing that the subset contains the zero matrix corresponds to a = b = 0. Showing that -U is in the subset corresponds to a = -1 and b = 0.