## Question

Let be defined by then,

1. Show that T is a linear transformation

2. What are the conditions on a, b, c such that (a, b, c) is in the null space of T? Specifically, find the nullity of T.

===========================================================================================================================================================

## Solution

Rewriting the transformation function T for convenience:

### Part 1 – Show that T is a linear transformation

We say that T is a linear transformation if it satisfies the below conditions:

(1)

(2)

__Proving #1__

(3)

Applying (3) in (1),

(4)

Similarly,

(5)

*Since (4) = (5), T satisfies the first condition.*

__Proving #2__

(6)

(7)

Similarly,

(8)

*Since (7) = (8), T satisfies the second condition as well.*

*Hence, T(u,v,w)=(u-v+2w,2u+v,-u-2v+2w) is a linear transformation.*

### Part 2 – Finding Nullspace & Nullity

Convert the linear transformation into the form, . Thus,

To find the nullspace, N(A), we need to apply Gauss Elimination to convert the matrix into its row-echelon form.

This can be done by applying in sequence:

Thus, the REF form of A,

& are equivalent systems.

Hence,

Observations from the above system:

- c is the free variable
- rank of the system is 2
- number of columns in the system is 3

Applying back-substitution

Applying Rank Nullity Theorem:

Rank(A) + Nullity(A) = No. of Columns = n

*Nullity(A) = 3-2 = 1 *