# Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem...   Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem 1, Section 4.3 and Problem 1, Section 4.6 that W is a subspace of P3. Let T:W+Pbe given by T(p(t)) = p' (t). It is easy to check that T is a linear transformation. (a) Find a basis for and the dimension of Range T. (b) Find Ker T, a basis for Ker T and dim KerT. (c) Is T one-to-one? Explain. (d) Is T onto? Explain. (e) Is T invertible? Explain.
Problem 5: Let V and W be vector spaces and let B = {V1, V2, ..., Un} CV be a basis for V. Let L: V → W be a linear transformation, and let Ker L = {ZEV: L(x)=0}. (a) If Ker L = {0}, show that C = { L(vi), L(02), ..., L(Un) } C W is a linearly independent set in W. (b) If C = {L(vi), L(v2),..., L(Un) } CW is a linearly independent set in W, show that Ker L = {0}. (Hint: Any z e KerL can be written uniquely as a linear combination of the basis vectors.) (c) If B = {V1, V2,..., Uk} CV is a linearly independent set in V, must C = {L(v1), L(v2),..., L(Uk) } CW be a linearly independent set in W?
(bil b2 b3 Problem 24 : Let b = E R4 be a fixed vector, b70. 64 Define L: R4 +R by L(x) = b 2, II 1 22 03 24 ER4 where b.x is the dot product of b and x in R4. (a) Show that L is a linear transformation. (b) Find the standard matrix representation of L. (c) Find a basis for and the dimension of Ker L. (d) Is L one-to one? Explain why. (e) Is L onto? Explain why. (f) Find a vector ve R4 such that v ¢ Ker L. (g) Find the nulity and the rank of L.

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• ### Problem 5: Let V and W be vector spaces and let B = {V1, V2, ...,... Problem 5: Let V and W be vector spaces and let B = {V1, V2, ..., Un} CV be a basis for V. Let L :V + W be a linear transformation, and let Ker L = {2 € V: L(x)=0}. (a) If Ker L = {0}, show that C = {L(v1), L(02), ..., L(vn) } CW is a linearly independent set in W. (b) If C = {L(01), L(V2),..., L(Un)} C W is a linearly independent set in W,...

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