d. (2 points) Calculate the image of T (i.e., Im(T)). You must show every step of the mathematical work that supports your conclusion to receive credit. e. (2 points) Sketch and label the image of 7 on the diagram below. T f. (2 points) Is T invertible? Why or why not? Justify your response using complete sentences. g. (2 points) Consider the set of vectors that is formed by a basis for the image of T and the Is the set linearly independent? Why or why not? Justify your response using complete sentences. h. (2 points) True or false: Let A be a 2 x2 matrix and 7 the linear transformation induced by A. If A has exactly one column of zeros, then the dimension of the kernel of T is 1 (i.e., dim(Ker(T)) = 1). e. (4 points) Prove your claim in part h. (Note that a "good" proof should communicate your logical argument. This means that you should use complete sentences in your proof so that there is no "guessing" what you mean! And it should include all the logical connections.)(22 points) Problem 4. This problem will ask you to sketch multiple objects on given diagrams. If you are handwriting your exam then please draw the following diagrams, including grid marks for each part that asks you to sketch If you are typing your exam then please take screenshot of the diagram and add your sketch to each part that asks for it. You can do this in a Jamboard that is not shared with anyone else or use Word's drawing features. Your response to this problem should include a total of three input-output diagrams. Consider A = . Let 7: 13 - 1? the linear transformation induced by the matrix A (1.e., T(x) = Ax). a. (2 points) Sketch and label the following vectors on the diagram below: e1: e2: T(e1). and 7(e2). (No credit will be given to sketches that do not have a label.) R2 T b. (2 points) Calculate the kernel of T (i.e., ker(7)). You must show every step of the mathematical work that supports your conclusion to receive credit. c. (2 points) Sketch and label the kernel of I on the diagram below. T