E231 HW 2, due Friday, September 23, 5:00 PMThis is kind of long. You’ll thank me later...1. If A is m × n, and m < n, show that A cannot have a left inverse. Similarly, if Ais m × n, and m > n, show that A cannot have a right inverse.2. Let M be an n × n matrix of integers. Assume det(M) 6= 0, so M is invertible(when viewed as a matrix of real, or complex, or rational numbers). Show thatM−1is itself a matrix of integers if and only if det(M ) = ±1.3. Let Sn×nbe defined asSn×n:=nA ∈ Rn×n: A = ATo(a) Show that Sn×n(along with the field R) is a vector space.(b) Find a basis for the space.(c) What is the dimension of the space?4. LetA =1 0 −1 1 00 1 1 1 1−1 1 0 1 1Find a basis for {x ∈ R5: Ax = 03}5. Suppose A ∈ Cn×n. Show that A is invertible if and only if x = 0n×1is the uniquesolution to Ax = 0n×1.6. Suppose (V, F) is a vector space, v1, v2∈ V , and the set {v1, v2} is a linearlyindependent set. For some a, b, c, d ∈ F, define w1:= av1+ bv2, w2:= cv1+ dv2.Show that {w1, w2} is a linearly independent set if and only if ad 6= bc.7. Let (V, F) be a finite dimensional space. Suppose {v1, v2, . . . , vn} be a basis for V .Let A ∈ Fn×n. As usual, let aijdenote the (i, j) element. Define vectors {ui}ni=1by the relationui:=nXj=1aijvjShow that {u1, u2, . . . , un} is a basis for V if and only if A is invertible.8. Let V be the set of polynomials with real coefficients of degree at most 3.(a) Convince yourself that this is a vector space (over the field R)(b) Using t as the indeterminate variable, show that {1, t, t2, t3} is a basis for V .(c) Show that {1, t − t2, t + 2t2, t3− t + 1} is a basis for V .1(d) Define a map A, mapping V to V by the ruleA(v) :=ddt(tv)Show that the map is a linear map.(e) Find the matrix representation of A with respect to the basis choice in part(8b).(f) Find the matrix representation of A with respect to the basis choice in part(8c).(g) Calculate the determinants of both representations of A.9. Using the change-of-basis formula, show that if A ∈ L(V, V ), where V is a finitedimensional vector space, then for any two matrix representations of A, denotedA1and A2, the determinants satisfydet A1= det A210. Let A, B, C be matrices of appropriate size. In each case, assume that X is re-stricted to be a matrix of appropriate dimension so that the expression is valid.Which of the following maps are linear ?(a) f (X) := AX + XB(b) f (X) := AX + BXC(c) f(X) := AX + XBX(d) f (X) := A0XA − X(e) f(X) := tr(AX)11. Let (V, F) be a finite dimensional vector space, and A ∈ L(V, V ). Assume that Ais invertible, and denote its inverse as calA−1. Let {v1, v2, . . . , vn} be a basis for V .Associated with any B ∈ L(V, V ), use M(B) to denote the matrix representationof B in this basis. Show that (as matrices)[M(A)]−1= M(A−1)12. (V, F) is a vector space, and R, T, S ⊂ V are subspaces. If S ⊂ R, show thatR ∩ (S + T ) (R ∩ S) + (R ∩ T )13. Suppose (V, F) and (W, F) are vector spaces, and A ∈ L(V, W ). S ⊂ W is asubspace. Define the inverse image of S by A as{v ∈ V : A(v) ∈ S} ⊂ VThis is denoted A−1(S), but is not to be confused with the inverse of A (since Amay not be invertible).2(a) Show that this is a subspace of V(b) If V and W are finite dimensional, showdimA−1(S)= dim (KerA) + dim (S ∩
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