Preparation problems for the discussion sections on October 7th and 9th1. Determine a basis for each of the following subspaces:(i) H = {4s−3s−t: s, t ∈ R},(ii) K = {abcd: a − 3b + c = 0},(iii) Col (1 2 3 0 00 0 1 0 10 0 0 1 0),(iv) Nul(1 2 3 0 00 0 1 0 10 0 0 1 0).2. Determine the dimension of Nul(A) and Col(A), whereA =1 2 3 −4 81 2 0 2 82 4 −3 10 93 6 0 6 9.3. Let A, B be two 4 × 3 matrices. Let a1, a2, a3be the columns of A and let b1, b2, b3be thecolumns of B.(i) Suppose that {a1, a2, a3} is linearly independent. Find a basis for Col(A) and describeNul(A).(ii) Suppose that {b1, b2} is linearly independent and b3= 2b1+ 7b2. Find a basis forCol (B) and a basis for Nul (B).4. Let u1=11, u2=1−1and let B = {u1, u2}.(i) Let v =23. Express v in terms of the basis B.(ii) Let w =11. Express w in terms of the basis B.(iii) Let T : R2→ R2be defined such that T (v) is expressing v in terms of the basis B.(Convince yourself that this is a linear transformation.) Determine the matrix thatrepresents T with respect to the standard basis of R2.15. Let L: R2→ R3be a linear transformation such thatL10=284, L01=301.What is L21?6. Let T : R2→ R3be the linear transformations withT1−1=501, T11=010.(i) Consider the basis B1= {10,01} of R2and the basis B2= {100,010,001} of R3.Determine the matrix A which represents T with respect to the bases B1and B2. Doyou have T (x) = Ax for all x ∈ R2?(ii) Consider the basis C1:= {1−1,11} of R2and the basis C2= {501,010,001} ofR3. Determine the matrix B which represents T with respect to the bases C1and C2.Do you have T (x) = Bx for all x ∈ R2?7. Let I : P3→ P4be the integration linear transformation that maps p toZt0p(t)dt.Consider the basis B = {1, t, t2, t3} of P3and the basis C = {1, t, t2, t3, t4} of P4. Determinethe matrix which represents I with respect to the bases B and
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