Construct a 2 × 2 matrix A such that the solution set of the equationAx = 0 is the line in R2 through (4, 1) and the origin. Then, find a vectorb in R2 such that the solution set of Ax = b is not a line in R2 parallelto the solution set of Ax = 0.

Context: [Linear Algebra] 6. For which conditions on b = (b1, b2, b3, b4) do there exist solution(s) for the linear system Ax = b? (a) A = [ 2 4 1 8 / 4 8 2 10 / 1 2 0 3 / 8 10 4 14] (b) A = [2 2 2 2 2 2 / 4 4 2 4 6 6 / 1 2 2 0 1 1 / 8 8 8 8 8 8]. Note that (a) is a 4x4 matrix and (b) is a 4x6 matrix. Please answer in concise format that's easy to read.

Suppose A is a 3 × 3 matrix and y is a vector in R3 such that the equationAx = y does not have a solution. Does there exist a vector z in R3 suchthat the equation Ax = z has a unique solution? Why?

Find the equation of the line that passes through the point (1, 4) and is parallel to the line

[Linear Algebra] 8. Let A = [2 1 6 1 / 3 1 7 2 / 4 1 8 3 / 5 1 9 3] (a 4x4 matrix), b = [1 1 k 4] (a 4x1 matrix). (a) Find condition on k ∈ R such that Ax = b, x ∈ R4 is solvable. (b) When the solution exists, find all solutions.

Vectors A and B lie in the x-y plane. We can deduce that A = B if:Group of answer choicesAx + Ay = Bx + ByAx = Bx and Ay = ByAx2 + Ay2 = Bx2 + By2Ax = Ay and Bx = ByAy /Ax = By /Bx