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.
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:
Applying (3) in (1),
Since (4) = (5), T satisfies the first condition.
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.
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 Rank Nullity Theorem:
Rank(A) + Nullity(A) = No. of Columns = n
Nullity(A) = 3-2 = 1