Problem 10. Let T : P3 → ×ばつ2 be the function defined by T(p) = [p(0) - p(1) ''(2)] . (a) Prove that T is a linear transformation. (b) Find the image of T. (c) Find the kernel of T. Rank-nullity theorem p(0)+p(1) Problem 11. In the exercises of the previous subsection, whenever you had computed one of the kernel/image of the transformation, could you have guessed the dimension of the other? Problem 12. Find all the surjective linear transformations from R3 to R4. Problem 13. Let V, W be finite-dimensional vector spaces. Prove that there exists an injective linear trans- formation L: VW dim(V) ≤ dim(W). Problem 5. (a) Consider a reflection R through the line y = x in R2. Find a basis of R2 for which it is easy to describe how R transforms the vectors inside it. Express R as a matrix transformation. (b) Consider a projection P on the line and 固 У = 2x in R2. Draw a picture of the line together with the vectors Describe how P transforms the two vectors. Can you express P as a matrix transformation? Problem 6. Let R denote a counterclockwise rotation by an angle of 0 in R2. Express R as a matrix transfor- mation.

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