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.

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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