1. (10 marks) Mark each statement true or false. Justify each answer: for true s

Question

1. (10 marks) Mark each statement true or false. Justify each answer: for true statements briefly quote relevant facts, for false ones, provide a counterexample. (a) If A is a 3 × 2 matrix, then the transformation Ar cannot map R2 onto R3 t are linearly dependent. (c) The equation Ax = b is homoegeneous if the zero vector is a solution (d) If A is an m × n matrix with m pivot columns, then the linear transformation A is a one-to-one mapping. (e) If a system of linear equations has no free variables, then it has a unique solution.

Explanation / Answer

a)True, since the transformation maps from R^2 to R^3 and 2<3, it can be one to one but not onto

b)True,conversely if x and y are linearly independent ,it must be that z can be written as linear combination of the other two, thus in their span

c)true, if zero is the solution then b=Ax =A(0) =0, SO Ax = 0 then homogenous solution

d)True , proved above

d) true, If a system has no free variables then has unique solution and if it has two differnt solution then will have infinte solutions

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