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2. For finite-dimensional vector spaces V and W over field F, let φ : V →...

2. For finite-dimensional vector spaces V and W over field F, let φ : V → W be a linear transformation.

A) If φ : R^83 → M(7 × 9, R) is surjective, what is dim(ker(φ))?

B) If φ : P24(R) → P56(R) is injective, what is dim(im(φ))?

C) If φ : M(40 × 3, R) → R^71 has dim(im(φ)) = 67, what is dim(ker(φ))?

D) If φ : R^92 → P105(R) has dim(ker(φ)) = 8, what is dim(im(φ))?

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