TBA 12 Hi guys and gals I need help with these Linear Algebr

TBA 1-2 ++++++++++++++Hi guys and gals. I need help with these Linear Algebra questions. They are true or false, and I would like to get an opinion from an expert. Thank you. A clear and correct answer will get a thumbs up guaranteed. Thank you!!!!!!!!!!!!!!++++++++++++++++++

Mark each statement true or false and justify your answer. No justification, no credit. For questions that are not truel false, provide clear and complete solutions. Every linear transformation from R\" to R\" is a matrix transformation. A mapping T :R\" R is one-to-one if each vector in R\" maps onto a unique vector in Rm 8. Why is the question \"Is the linear transformation T onto?\" an existence question? Explain clearly and completely

Solution

6) True.

If T(x) = Ax where A is a matrix then T(u+v) = A(u+v) = Au +Av and

T(cu) = A(cu) = c(Au)

Hence T is linear by matrix transformation.

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7) True.

Since T(u) = T(v) implies u =v we get T(u-v) =0

or T(0) =0

Hence T(x) =0 has only trivial solution and hence T is one to one.

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8) Suppose T: R^n to R^m is onto then it must necessarily follow that columns of A span R^m.

If columns of A span R^m, then we have the column vectors as the basis of R^m.

Thus proved.