M136 S21 – OPTIONAL Crowdmark Written Assignment 10
SUBMISSION DEADLINE IS THURSDAY AUGUST 5 AT NOON (EDT)
计算机代写 Material for CA-10 covers 19A , 19B, 19C, 20, 21A, 21B, 22, and any material from previous topics.CA-10 is an optional assignment so if ···
计算机代写
Material for CA-10 covers 19A , 19B, 19C, 20, 21A, 21B, 22, and any material from previous topics.
CA-10 is an optional assignment so if you do not submit it then the assignments CA-2 to CA-9 will count for 10% of the final grade (i.e. 1.25 mark each assignment), and if you do
submit CA-10, then the best 8 marks of your assignments CA-2 to CA-10 inclusively will count for 10% of your final grade ( i.e. 1.25 marks each as your lowest assignment mark will be dropped).
Fully justify your answers. Lemma numbers and Topic numbers must be quoted.
Question 1 [16 marks] 计算机代写
a.DiagonalizeA, that is, find an invertible matrix P and a diagonal matrix D such that
P−1AP = D.
b.Provethat A is invertible and that the inverse matrix of A is
c.ComputeA2301, A−1000, A−2301, where for any invertible matrix C, we define
Cn = CC . . . C(n factors) and C−n = (Cn)−1 = C−1C−1 . . . C−1(n factors).
Question 2 [5 marks] 计算机代写
Question 3 [6 marks]
Prove or disprove that S is linearly independent for all k R. Use the usual addition and scalar multiplication in M2x2(R).
Question 4 [6 marks] 计算机代写
Let p1, p2, p3 be vectors in P2(R), where
p1 = p1(x) = 1 + 2x + x2
p2 = p2(x) = 2 + 9x
p3 = p3(x) = 3 + 3x + 4x2
Use the usual addition and scalar multiplication in P2(R) to show that S = p1, p2, p3 form a basis for P2(R).
Question 5 [4 marks] 计算机代写
Consider V = R2 and define addition and scalar multiplication as follows: if u = (u1, u2)T , v = (v1, v2)T , and k ∈ R, then define
u ⊕ v = (u1 + v1, u2 + v2)T and k ⊙ u = (ku1, 0)T .
Determine whether (V, ⊕, R, ⊙) is a vector space.
Question 6 [ 4 marks]
Find a basis for the rowspace of A