I have a Linear Algebra midterm tomorrow and I\'d like a walkthrough for solving

Question

I have a Linear Algebra midterm tomorrow and I'd like a walkthrough for solving these problems. I'll provide the solutions, but again I'd like some explanations. Thanks for your help! *To prevent confusion: A is a matrix with 3 rows and 5 columns* Let A = [(1, 2, 3, 0, 1), (3, 6, 4, 4, 6), (2, 4, 1, 5, -3)]. Then RREF(A) = [(1, 2, 0, 0, 10), (0, 0, 1, 0, -3), (0, 0, 0, 1, -4)]. (i) What is the rank of A? 3. (ii) What is the dim of Row(A)? 3. (iii) What is dim of Col(A)? 3. (iv) What is dim of Nul(A)? 2. (v) Give a basis for Row(A). The set consisting of the first three rows of RREF(A). (vi) Give a basis for Col(A). Basis for Col(A) = {(1 3 2)^T, (3 4 1)^T, (0 4 5)^T}. (vii) Give a Basis for Nul(A). (Show Work) {(-2 1 0 0 0), (-10 0 3 4 1)} = {u, v} (viii) Give the General solution to Ax = [(0), (4), (5)]. No computation needed! x = au + bv + [(0), (0), (0), (1), (0)]

Explanation / Answer

the first row gives x1=-x3 the second gives x2=x3 so you can write (x1,x2,x3) = (-x3,x3,x3) = (-1,1,1)x3 then you introduce a dummy variable x3=t so the equation of the line is L = (-1,1,1)t for all t or x1=-t, x2=t, x3=t in parametric form As another example you might have A = [1,-2,0;0,1,-1;0,0,0] in that case you would get x1=2x2 x2=x3 so (x1,x2,x3) = (2x2,x3,x3) = (2,0,0)x2 + (0,1,1)x3 then you have to introduce 2 dummy variables to get P = (2s,t,t) = (2,0,0)s + (0,1,1)t as the equation of the plane or x1=2s, x2=t, x3=t in parametric form

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