Worksheet: matrix transformations

This worksheet deals with matrix transformations, and in particular, kernel and image. The goal is to understand these important subspaces in a familiar context.

Let

A

be an

m \times n

matrix. We can use

A

to define a transformation

T_{A} : R^{n} \to R^{m}

given by

T_{A} (x) = A x,

where we view

x

as an

n \times 1

column vector.

The kernel of

T_{A}

is the set of vectors

x

such that

T_{A} (x) = 0 .

That is,

\ker T_{A}

is the set of solutions to the homogeneous system

A x = 0 .

The image of

T_{A}

(also known as the range of

T_{A}

) is the set of vectors

y \in R^{m}

such that

y = A x

for some

x \in R^{n} .

In other words,

im (T_{A})

is the set of vectors

y

for which the non-homogeneous system

A x = y

is consistent.

Because

T_{A}

is a linear transformation, we can compute it as long as we’re given its values on a basis. If

{v_{1}, v_{2}, \dots, v_{n}}

is a basis for

R^{n},

then for any

x \in R^{n}

there exist unique scalars

c_{1}, c_{2}, \dots, c_{n}

such that

x = c_{1} v_{1} + c_{2} v_{2} + \dots + c_{n} v_{n},

and since

T_{A}

is linear, we have

T_{A} (x) = c_{1} T_{A} (v_{1}) + c_{2} T_{A} (v_{2}) + \dots + c_{n} T_{A} (v_{n}) .

The main challenge, computationally speaking, is that if our basis is not the standard basis, some effort will be required to write

x

in terms of the given basis.

1.

Confirm that

B = {[\begin{matrix} 1 \\ 0 \\ 2 \\ 3 \end{matrix}], [\begin{matrix} 4 \\ 2 \\ 0 \\ - 3 \end{matrix}], [\begin{matrix} 0 \\ 4 \\ - 3 \\ 2 \end{matrix}], [\begin{matrix} 3 \\ 5 \\ - 2 \\ 1 \end{matrix}]}

is a basis for

R^{4} .

To assist with solving this problem, a code cell is provided below. Recall that you can enter the matrix

[\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}]

as Matrix([[a,b,c],[d,e,f],[g,h,i]]) or as Matrix(3,3,[a,b,c,d,e,f,g,h,i]).

The reduced row-echeleon form of A is given by A.rref(). The product of matrices A and B is simply A*B. The inverse of a matrix A can be found using A.inv() or simply A**(-1).

One note of caution: in the HTML worksheet, if you don’t import sympy as your first line of code, you’ll instead use Sage syntax. Sage uses A.inverse() instead of A.inv().

In a Jupyter notebook, remember you can generate additional code cells by clicking on the + button.


    
        
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from sympy import Matrix,init_printing
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init_printing()

    
    
    
    
        
            
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You can also use the cell below to write down any necessary explanation.

2.

Write each of the standard basis vectors in terms of this basis.

Suggestion: in each case, this can be done by solving a matrix equation, using the inverse of an appropriate matrix.

A linear transformation

T : R^{4} \to R^{4}

is now defined as follows:

T ([\begin{matrix} 1 \\ 0 \\ 2 \\ 3 \end{matrix}]) = [\begin{matrix} 3 \\ 0 \\ 2 \\ - 1 \end{matrix}], T ([\begin{matrix} 4 \\ 2 \\ 0 \\ - 3 \end{matrix}]) = [\begin{matrix} 1 \\ 2 \\ 0 \\ 5 \end{matrix}], T ([\begin{matrix} 0 \\ 4 \\ - 3 \\ 2 \end{matrix}]) = [\begin{matrix} 4 \\ 2 \\ 2 \\ 4 \end{matrix}], T ([\begin{matrix} 3 \\ 5 \\ - 2 \\ 1 \end{matrix}]) = [\begin{matrix} 2 \\ 4 \\ 0 \\ 10 \end{matrix}] .

Let

{e_{1}, e_{2}, e_{3}, e_{4}}

denote the standard basis for

R^{4} .

3.

Determine

T (e_{i})

for

i = 1, 2, 3, 4,

and in so doing, determine the matrix

A

such that

T = T_{A} .

4.

Let

M

be the matrix whose columns are given by the values of

T

on the basis

B .

(This would be the matrix of

T

if

B

was actually the standard basis.) Let

N

be the matrix whose inverse you used to solve part (b). Can you find a way to combine these matrices to obtain the matrix

A ?

If so, explain why your result makes sense.

Next we will compute the kernel and image of the transformation from the previous exercises. Recall that when solving a homogeneous system

A x = 0,

we find the RREF of

A,

and any variables whose columns do not contain a leading 1 are assigned as parameters. We then express the general solution

x

in terms of those parameters.

The image of a matrix transformation

T_{A}

is also known as the column space of

A,

because the range of

T_{A}

is precisely the span of the columns of

A .

The RREF of

A

tells us which columns to keep: the columns of

A

that correspond to the columns in the RREF of

A

with a leading 1.

Let

T

be the linear transformation given in the previous exercises.

5.

Determine the kernel of

T .

6.

Determine the image of

T .

7.

The Dimension Theorem states that for a linear transformation

T : V \to W,

where

V

is finite-dimensional,

\dim V = \dim \ker (T) + \dim im (T) .

Confirm that your results on this worksheet agree with this theorem.

Linear Algebra: A second course, featuring proofs and Python

Worksheet 2.4 Worksheet: matrix transformations
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