Linear Algebra

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This question tests your understanding of linear transformations; in
particular, it tests your ability to determine the matrix of a linear
transformation with respect to given bases and to find the kernel and image
of a linear transformation.

The function t : R3 -? R3 is given by the rule
(x, y, z) 0-? (y – z, x + z, x + y).

(a) Use Strategy 1.1 in Unit LA4 to show that t is a linear transformation.
(b) Write down the matrix of t with respect to the standard basis for R3.
(c) Determine the matrix of t with respect to the basis
{(1, 0, 0), (1, 1, 0), (0, 1, 1)} for the domain and the standard basis for
the codomain.
(d) Find the kernel of t, describe it geometrically and state its dimension.
(e) Find a basis for the image of t, state the dimension of the image and
describe the image geometrically.
(f) Let s be the linear transformation
s : P3 -? R3
a + bx + cx2 0-? (a + c, b, a + b + c).
Find the matrix of s and the matrix of t ? s with respect to the
standard basis for the domain P3 and the standard basis for the
codomain R3.

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