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## Linear Algebra, Linear Transformations

it is easy to show that ax+ b= (4a- 7b)(9x+ 5)+ (9b-5a)(7x+ 4) so that T(ax+ b)= (4a- 7b)T(9x+5)+ (9b-5a)T(7x+4)= (4a+ 7b)(.1,.2)+ (9b-5a)(.3, .8)= (-1.1a+ 3.4b, -3.2a+ 8.6), a unique linear transformation.

Basically, where you stop explaining and start handwaiving (see above) is where I need to more fully explain.

Please do show how to arrive at the conclusion you mention above, as well as where you get the values from (like 4a and 7b), along with why you set it up the way you did (say, why did you multiply (4a -7b)(9x+5).. etc?