Properties - Linear Algebra - Solved Exam, Exams of Linear Algebra

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Math 205A Final Exam (100 points) Name: Sol Ww bo “AS « Check that you have 10 questions on five pages. « Show all your work to receive full credit for a problem. 1, (18 points) Short answers: (No explanations needed, Simply write your answers. If you do some calculation to get the answer, show the calculation.) (a) (6 points) Let A and B be 4x4 matrices, with det A = —4 and det B = 7. Use properties of determinants to compute: (i) det A? = _ I a | det A -4 (i) det (ABT = tek CAB) = Cet A) Cdet B) = C47 SY (b) Suppose C is a 4 x 3 matrix with a pivot in every column. (i) What is the dimension of Nul C? dim Nol CZ number of Fee vasebles=O - (ii) What is the dimension of Col C? dim Co} C= number of prot uo lumns= 3- c) Define a linear transformation T’ : R? + y T(x), t2) = (22, + v2, 21 — tg,X2). Fin ) Defi li fi ition T ; R? + R? by T( ) = (2a, + ). Find the standard matrix of T. 7(eeD)= [eee] = [2TH] [Ea] 5 shandsrd wots dF T = 2 4 } =| ol!