Linear_algebra.zip - (92.08 KB)**[**

Linear_algebra.docx**]**

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.