# Linear algebra and linearly independence

## Homework Statement

I have three vectors in R^(2x2):

(1 0 , 0 1) (That is "1 0" horizontal first line, and "0 1" horizontal second line), (0 1, 0 0) and (0 0, 1 0).

I have to determine if they are linear independent or not. I know how to do it in R^(2x1), but not in R^(2x2). What's the method?

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Homework Helper
Are these matrices?

Office_Shredder
Staff Emeritus
Gold Member
It's the same thing

If

a( (1,0),(0,1) ) + b( (0,1),(0,0) ) + c( (0,0), (1,0) ) = 0 (0 being ( (0,0),(0,0) )
then

( (a,b),(c,a) ) = ( (0,0),(0,0) ) so comparing positions, we get that a=b=c=0

(this assumes I interpreted R^(2x2) correctly, but I think I did)

Yes, these are matrices.

So the example is linearly independent, because they cannot be written as a linear combination?

If we take a new example: a(1,0)(0,1) + b(0,1)(0,0) + c(2,3)(0,2) = 0.

Then we get:

(a + 2c, b+3c)(0, a + 2c) = 0 so it is linearly dependent?

HallsofIvy
$$-2\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]- 3\left[\begin{array}{cc}0 & 1 \\ 0 & 0\end{array}\right]+ \left[\begin{array}{cc}2 & 3 \\0 & 2\end{array}\right]= \left[\begin{array}{cc}0 & 0 \\ 0 & 0 \end{array}\right]$$