Math 220 Assignment

Let f be a linear map from R^{3} to R^{3} for which
1. f(1, 0 ,0) = (1, 2, 3).
2. f(0, 1/2, 0) = (3, 2, 1).
3. f(-1, 0, 2) = (4, -6, 2).
Find all possible 3 \times 3 matrices A for which the formula f(x) = A x is valid for all x in R^{3}. Hint: Use the rules for abstract linearity to work out what happens under f to (0, 1, 0) and (0, 0, 1).
Let g be the linear map from R^{4} to R^{4} that is defined by g(x) = B x where B is the matrix

(

1

2

-4

3

-2

-1

-1

5

1

3

2

-1

1

1

-1

-1

)
.
Find a 4 \times 4 matrix C for which the linear map h given by multiplication by C has the property that both h(g(x)) = x and g(h(y)) = y for all x and all y in R^{4}.
Could the previous exercise have been completed successfully if the given matrix B had been one of the matrices appearing in the assignment due Friday, October 12?