The linear transformation T is defined by T(x)=Ax.Find (a) ker(T), (b) nullity(T), (c) range(T), (d) rank(T):-

Question

A= ⌈ 4 1 ⌉

0 0

⌊ 2 -3⌋

Answer 1

Let's start with (b) and (d) because they are the simplest. Nullity is simply the number of free columns and rank is the number of pivot columns. Put the matrix A in row echelon form and count the number of each type of column (remember that a pivot column is a column with a pivot and a free column is one without a pivot).

Now for the other two parts:

(c) The range of the transformation T is the set of all possible outputs. Try multiplying Ax where x is a some generic column vector like [x₁, x₂, x₃, ..., x_n]^T for however many rows you need. You should get an expression for a vector in terms of two parameters for this problem, something like this:

⌈ 1 ⌉ ⌈ 4 ⌉

| 2 | x₁ + [ 5 ]x₂

⌊ 3 ⌋ ⌊ 6 ⌋

(a) The kernel of the transformation T is the set of every vector x that you can input and get 0 as an output. This is the same as the null space of A and the solutions to the homogeneous system Ax = 0.

You can find the kernel just like you would find the homogeneous solution set. Remember that we are expecting a set of 0, 1, or infinity solutions. First, augment the matrix with a column of zeros to represent the equation Ax = 0

⌈ 4 1 | 0 ⌉

| 0 0 | 0 |

⌊ 2 -3 | 0 ⌋

Then reduce it to row echelon form like you did for parts (b) and (d). Once you've done that, write out the equations represented by the reduced matrix. For example, the row [ 2 9 | 0] would represent 2x₁ + 9x₂ = 0. If the equations are inconsistent the kernel is the empty set {}. If there are no free variables, solve the system to find the one vector that works. Otherwise, solve the equations to find each pivot variable in terms of the free variables, producing a parametric expression for the solution set like in part (c).