Prove Corollary 6.2. If L: V → W is a linear transformation of a vector space V into a vector space W and dim V=dim W, then the following statements are true: (a) If L is one-to-one, then it is onto. (b) If L is onto, then it is one-to-one.

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3. Which of the following pairs of vector spaces are isomorphic? Justify your answers. F³ and P3 (F). F4 and P3(F). (a) (b) (c) (d) M2x2 (R) and P3(R). V = {A € M₂x2 (R) : tr(A) = 0} and R4.

Let V be a vector space, v, u € V, and let T₁ : V → V and T₂ : V → V be linear transformations such that T₁(u) = −5v + 3u, T₁ (v) = 5v +2u, T₂ (v) = −3v - 2u, T₂(u) = 5v – 2u. Find the images of v and u under the composite of T₁ and T₂. (T₂T₁)(v) = = (T₂T₁)(u) =

Which of the following statements is (are) true if B is a basis for the vector space V (a) In some situations, a plane in R6 can be isomorphic to R4 (b) If x is in V and B contains n vectors, the the B- coordinate vector of x is in R (c) The vector spaces P5 and RS are isomorphic. (d) If PB is the change of coordinates matrix, then [x]B = PB x for x in V (e) A coordinate mapping is a bijective mapping O All of the above Noneoftheabove Options a, b, c and e Options a, b and e Options b, c and e Options a, b, d and e Options b, c, d and e Options a, and b

Proposition 1: Let T: V-> W be a linear transformation of linear spaces V and W, suppose that a set CW is linearly of vectors {v} →→ V1, V2, V3 }} CV has the property that T {T (v₁),T (√₂²),T (√₂)} (v independent. Then, the set of vectors {0₁, 02, 07 } Below is a "proof" of the above proposition. You may use it as a template and upload your proof below. → → → Proof: Suppose that a₁v₁ + a2v₂ + a3v3 Then, applying T to both sides we get Since T is a linear transformation, we get T Since {7 (5),(6). (6) {T T T = V3 CV is linearly independent. Ō. are linear independent we get that