## 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