How to do this question , in relations to transformation?

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1 Answer
George C.
Feb 2, 2018

#(-6, 7)#

Explanation:

I am not sure what your preferred convention is for this, but we can represent the linear transformation by right multiplication by a suitable #2 xx 2# matrix as follows:

Since we are given the images of the units #(1, 0)# and #(0, 1)# we can write our transformation as right multiplication by the matrix whose rows are the images of those points, namely:

#((2, -1), (-2, 5))#

So:

#(1, 0)((2, -1), (-2, 5)) = (2, -1)#

#(0, 1)((2, -1), (-2, 5)) = (-2, 5)#

Then:

#(-2, 1)((2, -1), (-2, 5)) = (-6, 7)#

Alternatively, we can use left multiplication of column vectors by a matrix like this:

#((2, -2), (-1, 5))((1), (0)) = ((2), (-1))#

#((2, -2), (-1, 5))((0), (1)) = ((-2), (5))#

#((2, -2), (-1, 5))((-2), (1)) = ((-6), (7))#

Using either convention, we find the image of #(-2, 1)# is #(-6, 7)#

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