# ZOOMLet V and W be vector spaces over a field K, and f: V -> W a linear map.(a) Let f R4 -+ R4 be a linear map defined by f(e) = ie; for 1 < i < 4 where(e1,.., e4) I, the basis consisting of unit vectors. Let1111Н —1-11E R4x41-11(a.1) Compute H'H and H. Deduce that the columns of H form a basis for R4(a.2) Find the dual basis of H. (Describe the linear forms explicitly.)(a.3 Find the matrix of f under the basis H of R4(b) Suppose V and W have the same fiite dimension n. Prove the following:(b.1) If f is injective, then for any basis u,..., u of V, the images f(u1),... ,f(1,)form a basis for W, hencef is onto;(b.2) If f is onto, then for any basis u, . .. , u Oof V, the images f(u1), ... ,f(u,) forma basis for W, hence f is injectiveRemark. Note that neither of above is true when the dimension is infinite (seeexamples in class). In your proof, you may use the fact that any set of independentvectors in a vector space can be extended to basis and any set of dependent vectorscontains a subset that is a basis.(c) Let X C V be a linear subspace such that X C Nullf), and let T: V -> V/X bethe natural surjection T(v) = 7.(c.1) Let f V/X > W be given by f(7)defined, that is, v, vi E V are such that u = v, (mod X), then (v) = T(v1), sof(7) is independent of the choice of representative of .) Prove that f is a lineartransformation such that foT f and f(V/X) = f(V)(c.2) Prove that if S: V/X - W is a a linear transformation such that S oT ff(v). (First shows that this is well-then S

Question

Hello, please I need a step by step and a well self-explanatory solution to questions (a.1), (a.2) and (a.3). I don't have a strong background in Algebra. HELP ME

Step 1

We are given that

Step 2

Again, we need to find the inverse of H, which is H-1 by Augmented matrix method.

Step 3

Continuing fu...

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