Determining if Set of Vectors Spans M2,2

[yanjt]

Hi,I have a question like this :
Determine whether the given set of vector spans M_2,2
{(5 3;0 0),(0 0;5 3),(6 -1;0 0),(0 0;6 1)}

I wonder can I jus directly prove that
a(5 3;0 0)+b(0 0;5 3)+c(6 -1;0 0)+d(0 0;6 1) = (5a+6c 3a-c;5b+6d 3b+d)
is also a M_2,2 and hence,it is a vector sets spans M_2,2?

Thanks a lot!

 

[matt grime]

The definition of a set of vectors spanning the whole of the vectors space is what? Because you've not stated it, nor verified it is true.

Note that this being M_2,2, is irrelevant: it is just a 4-d vector space, so there's no need to write things as matrices.

 

[HallsofIvy]

Hi,I have a question like this :
Determine whether the given set of vector spans M_2,2
{(5 3;0 0),(0 0;5 3),(6 -1;0 0),(0 0;6 1)}

I wonder can I jus directly prove that
a(5 3;0 0)+b(0 0;5 3)+c(6 -1;0 0)+d(0 0;6 1) = (5a+6c 3a-c;5b+6d 3b+d)
is also a M_2,2 and hence,it is a vector sets spans M_2,2?

Thanks a lot!

No, the fact that the particular linear combination is in M_2,2 only means that these matrices span some subspace of M_2,2- which is true of any collection of matrices in M_2,2. What you must prove is that any matrix in M_2,2 can be written as a linear combination of them.

(u v; x y) is in M_2,2 for any real numbers u, v, x, y. Can you find a, b, c, d such that a(5 3;0 0)+b(0 0;5 3)+c(6 -1;0 0)+d(0 0;6 1)= (u v; x y) for any u, v, x, y?

Also, since, as Matt Grime said, this is a 4 dimensional space and your set contains exactly 4 matrices, this set spans M_2,2 if and only if it is independent. If a(5 3;0 0)+b(0 0;5 3)+c(6 -1;0 0)+d(0 0;6 1)= (0 0; 0 0), what must a, b, c, d be?