2. The transpose of a linear function f : R" - R" is a linear function g : Rm -> R" such that f(v) w = v g(w) for all v E R" and we Rm. Provide answers with justification to the following questions: (a) If A is the standard matrix of f, then what is the standard matrix of the transpose of f? (First try this for a specific matrix A like the one in problem #1. Then find a general solution.) (b) Why is there only one transpose of f? (In other words, suppose h : Rm -> R" is another transpose, and then explain why g = h.)Certainly! Let's follow the instructions in the question by first working with a specific matrix A, then generalizing our findings. Part (a): If A is the standard matrix of f, what is the standard matrix of the transpose of f? Step 1: Consider a specific matrix A. Let's take a simple example. Suppose f : R2 - R2 and the standard matrix of f is: A = This means for any vector v = V1 U2 E R', the transformation f (v) is given by: f(v) = Av = v1 + 202 = 3v1 + 4v2 Step 2: Compute the transpose of f. To find the transpose of f, we need a new linear function g : R" - R such that for any vectors v, WE R?: f (v ) w = vg(w)V1 W1 Using our matrix A, for v = and w = U 2 W2 f (v) = v1 + 202 3v1 + 4v2 f (v ) w = v1 + 202 W1 3v1 + 402 W2 = (21 + 202)w1 + (301 + 402)202 Now, for the transpose function g(w) = ATw, we have: W1 + 3w2 g(w) = A w = W1 NH W2 2w1 + 4w2 Let's check if this satisfies the definition of the transpose: u g(w) = W1 + 3w2 = v1(w1 + 3w2) + 12(2w1 + 4w2) V2 2w1 + 4w2 Simplifying: viw1 + 3v1w2 + 202w1 + 4v2w2 = (v1 + 202)w1 + (3v1 + 4v2)w2 This matches f(v) w, so the transpose function g(w) is indeed defined by the matrix A = 3 N 4 Step 3: Generalize the result. In general, if f : " -> RM has a standard matrix A, then the transpose of f is a function g : pm - R" whose standard matrix is AT.Part (b): Why is there only one transpose of [? Now, suppose there exists another transpose h : R R". This function would also satisfy: f(v) w=v"h(w) forallv R" and w R. However, we have already shown that the matrix representing the transpose must be A" . Since matrices representing linear transformations are unique, h(w) must be the same as g(w). Thus, h = g, and there is only one transpose of f. This concludes the justification for why there is only one transpose