16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.1.7

Linear Transformations

Mathematics — Linear Algebra · range-null-space-rank-nullity-explicit

Problem

Let

F

be a subfield of the complex numbers and let

T

be the function from

F^3

into

F^3

defined by

T(x_1, x_2, x_3) = (x_1 - x_2 + 2x_3,\; 2x_1 + x_2,\; -x_1 - 2x_2 + 2x_3).

(a) Verify that

T

is a linear transformation. (b) If

(a, b, c)

is a vector in

F^3

, what are the conditions on

a

b

, and

c

that the vector be in the range of

T

? What is the rank of

T

? (c) What are the conditions on

a

b

, and

c

that

(a, b, c)

be in the null space of

T

? What is the nullity of

T