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1. Let V and W be finite-dimensional vector spaces over field F with dim(V) = n...

1. Let V and W be finite-dimensional vector spaces over field F with dim(V) = n and dim(W) = m, and

let φ : V → W be a linear transformation.

A) If m = n and ker(φ) = (0), what is im(φ)?

B) If ker(φ) = V, what is im(φ)?

C) If φ is surjective, what is im(φ)?

D) If φ is surjective, what is dim(ker(φ))?

E) If m = n and φ is surjective, what is ker(φ)?

F) If m = n and im(φ) = W, what is ker(φ)?

G) If im(φ) = {0}, what is ker(φ)?

H) If φ is injective, what is ker(φ)?

I) If φ is injective, what is dim(im(φ))?

J) If m = n and φ is injective, what is im(φ)?

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