Can Four Specific 2x2 Matrices Span M22?

[tua96426]

Homework Statement

Is M22 spanned by the following 2x2 matrices:

[1,1;0,1]
[0,1;1,0]
[1,0;1,1]
[0,-1;1,0]

semi colon means start of new row.

Homework Equations

The Attempt at a Solution

Is this essentially asking me to check if the given matrices form a spanning set? So are we checking for linear independence? Or are we saying that ok let's pick an arbitrary 2x2 [a,b;c,d] and see if we can come up with a formula for a b c and d in terms of 4 scalars?

I checked for linear independence, and turns out that these 4 matrices are not linearly independent. If they were than after row reduction, I should have gotten I.

Any help on this would be appreciated.

 

[Mark44]

Homework Statement

Is M22 spanned by the following 2x2 matrices:

[1,1;0,1]
[0,1;1,0]
[1,0;1,1]
[0,-1;1,0]

semi colon means start of new row.

Homework Equations

The Attempt at a Solution

Is this essentially asking me to check if the given matrices form a spanning set?

Yes, that's exactly what it's asking.

So are we checking for linear independence? Or are we saying that ok let's pick an arbitrary 2x2 [a,b;c,d] and see if we can come up with a formula for a b c and d in terms of 4 scalars?

If the four matrices span M₂₂, they will necessarily be linearly independent. If you show that they are linearly independent, they will necessarily have to span M₂₂. Either condition implies the other in this problem.

I checked for linear independence, and turns out that these 4 matrices are not linearly independent. If they were than after row reduction, I should have gotten I.

I haven't checked that the four matrices are linearly independent, so I can't say. I don't understand what it is that you row reduced, or how it is that something should have reduced to the identity matrix.

If you checked linear independence you would be solving the equation a*M₁ + b*M₂ + c*M₃ + d*M₄ = 0, where the M_is are your four matrices, and where 0 is the 2 x 2 zero matrix. If the matrices are linearly independent there will be just one solution for the constants; namely, a = b = c = d = 0. If the matrices are linearly dependent, there will be multiple solutions for the constants.

If you check to see whether the matrices span M₂₂, you'll be solving the equation a*M₁ + b*M₂ + c*M₃ + d*M₄ = A, where A is an arbitrary 2 x 2 matrix.

Any help on this would be appreciated.

 

[tua96426]

Thanks Mark.
I didn't know how to describe the matrix that I was row reducing.. but yea that's exactly what I meant.
Thanks a lot, it cleared a lot of ambiguities.

-Sudhi