Question 1 Matrices as Linear Transformations Let A: R3 R4 be the linear transformation (Links) mapping the 3-vector in the input layer to the 4-vector in the hidden layer, and let B: R4 → R² be the linear transformation (Links) from the hidden layer to the output layer. Write the matrix representations of A and B, and label each edge in the image below with the corresponding entries of A and B. Input x1 Hidden nodes layer Input nodes layer Output nodes layer Output y1 Input x2 Output y2 Input x3 Neuron Links Links Question 2 Change of Basis between standard coordinates in R³ Sometimes, preprocessing of raw data is useful before training a neural net- work. Perhaps we know something about the structure of our feature data and want to put it into a more convenient basis before training. For example, if we know that the data is cylindrically symmetric, it might make sense to train our algorithm based on distance to the center of the cylinder, height on the cylinder, and angle from the center, rather than the x, y, and z distance from the origin. a. Write the change of basis matrix P : R³ →R³ which transforms the standard basis E = {₁, j, k} to a new basis B, where B = {2 0 } b. The transformation from Cartesian coordinates (x1, x2, x3) to cylindrical coordinates (p, o, x3) is given by the following transformation T: R³ → R³ p = √x² + x²², 4 = -1 x2 = tan ' x3 = x3. x1 Is T linear? If yes prove it. If not, give a counter example. c. Consider the following matrix (the Jacobian matrix of this transformation) θρ θρ ap Jx3 Jx1 8x2 do მი მი J = მე1 ǝx2 ǝx3 8x3 ǝx3 Jx3 Show that, if p R² → R and მე1 მ2 Əx3 : R² → R were replaced with linear functions, then the J would be the matrix representation of the T. (Since T is not linear, J can be thought of as a linear approximation of T).