Find the image of an arbitrary vector

Question

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Answer 1

We know that the linear transformation T takes in a 2-D vector and returns a transformed image of a 2-D vector. This means that it can be written as a 2x2 matrix, where the transformation of a vector v is given by v' = Tv (i.e. matrix multiplication). We also know how T transforms two given vectors, so we can write the matrix in terms of unknown variables that can be solved for:

T = [(a, b) (c, d)] where (a, b) is the first row and (c, d) is the second.

Then, we can write the two given transformations in terms of these unknown coefficients and solve:

First vector:

T(v1) = [(a, b) (c, d)][(-3, 4)] (right hand side)

T(v1) = [(-32, 11)] (left hand side)

Simplifying the right hand side using the rules for how a 2x2 matrix acts on a 2-d vector:

T(v1) = [(-3a + 4b), (-3c + 4d)]

Now we can equate the left and right hand sides to give us two equations:

-3a + 4b = -32

-3c + 4d = 11

Similarly for the second vector:

T(v2) = [(a, b) (c, d)][(2, -3)] (right hand side)

T(v2) = [(24, 8)] (left hand side)

T(v2) = [(2a - 3b), (2c -3d)]

Giving us two more equations:

2a - 3b = 24

2c - 3d = 8

So we have 4 equations with 4 unknowns a, b, c, and d. Solve this system using your favorite method, and then to find the image of an arbitrary vector [(x, y)] you can use the same transformation rule:

T(v) = [(a, b) (c, d)][(x, y)] = [(ax + by), (cx + dy)] (except using the values for a,b,c,d you got from solving the system of 4 equations)