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## Homework Statement

Let ##\mathbb{F}_3## denote the field with 3 elements and let ##V = \mathbb{F}_3^2##. Let ##\alpha, \beta, \gamma, \delta## be the four one-dimensional subspaces of ##V## spanned by ##(1,0), (0,1), (1,1)## and ##(1,-1)## respectively. Let ##\operatorname{GL}_2 (\mathbb{F}_3)## act on ##\{ \alpha, \beta, \gamma, \delta \}## by matrix multiplication.

Find the kernel of the homomorphism corresponding to this action.

## Homework Equations

## The Attempt at a Solution

So I don't think this problem will be very difficult, but I have a confusion. How exactly does the general linear group act on those subspaces? If they were particular elements, it would be clear how it acts on them by matrix multiplication. But I'm not seeing how this is done when they are subspaces.