Matrices acting on position vectors | Year 11

01 / LEARN

Understand the idea.

Images of the basis vectors

The standard unit vectors select the columns: $A e_{1}$ is the first column and $A e_{2}$ the second. Since $p = x e_{1} + y e_{2}$ , linearity gives $A p = x A e_{1} + y A e_{2}$ , a weighted sum of those column images.

To construct a linear map from unit-vector images, place the image of $e_{1}$ in column 1 and the image of $e_{2}$ in column 2. This determines every output, not only the two stated images.

Map every vertex consistently

Transform a polygon by applying the same matrix to each vertex, then joining the images in the original order. Straight edges remain straight or collapse to points, although lengths and angles may change.

The retained diagram shows the map $(x, y) \mapsto (x + y, y)$ . The added horizontal displacement depends on height, so this is not a translation. Calculate the vertex images to check the sketch rather than relying on appearance alone.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Read the columns

For $A = (\begin{array}{cccccccccc} 2 & - 1 \\ 1 & 3 \end{array})$ , find $A e_{1}$ , $A e_{2}$ and the image of $(2, 1)$ .

Worked solution

1
Read the first unit-vector image.
$A e_{1} = (\begin{array}{cccccccccc} 2 \\ 1 \end{array})$
Multiplication by ${(1, 0)}^{T}$ selects column 1.
2
Read the second unit-vector image.
$A e_{2} = (\begin{array}{cccccccccc} - 1 \\ 3 \end{array})$
Multiplication by ${(0, 1)}^{T}$ selects column 2.
3
Express the point using the unit vectors.
$(\begin{array}{cccccccccc} 2 \\ 1 \end{array}) = 2 e_{1} + e_{2}$
The input coordinates specify the weights of the two basis vectors.
4
Apply the same weights to their images.
$\begin{aligned} A (\begin{array}{cccccccccc} 2 \\ 1 \end{array}) & = 2 (\begin{array}{cccccccccc} 2 \\ 1 \end{array}) + (\begin{array}{cccccccccc} - 1 \\ 3 \end{array}) \\ = (\begin{array}{cccccccccc} 4 - 1 \\ 2 + 3 \end{array}) \\ = (\begin{array}{cccccccccc} 3 \\ 5 \end{array}) \end{aligned}$
Linearity distributes the transformation across the weighted sum.

Answer

Unit-vector images $(2, 1)$ and $(- 1, 3)$ ; point image $(3, 5)$ .

EXAMPLE 02

Construct a matrix

A linear map sends $(1, 0)$ to $(0, 2)$ and $(0, 1)$ to $(- 3, 0)$ . Find its matrix and the image of $(1, 2)$ .

Worked solution

1
Put the given images into columns.
$A = (\begin{array}{cccccccccc} 0 & - 3 \\ 2 & 0 \end{array})$
The first column is the image of $(1, 0)$ ; the second is the image of $(0, 1)$ .
2
Multiply by the requested input.
$A (\begin{array}{cccccccccc} 1 \\ 2 \end{array}) = (\begin{array}{cccccccccc} 0 (1) + (- 3) (2) \\ 2 (1) + 0 (2) \end{array})$
Each output coordinate is a row-column dot product.
3
Simplify the image coordinates.
$A (\begin{array}{cccccccccc} 1 \\ 2 \end{array}) = (\begin{array}{cccccccccc} - 6 \\ 2 \end{array})$
The two basis images contribute once and twice respectively.

Answer

$A = (\begin{array}{cccccccccc} 0 & - 3 \\ 2 & 0 \end{array})$ , image $(- 6, 2)$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

What is the image of $(0, 1)$ under $(\begin{array}{cccccccccc} 3 & 4 \\ - 1 & 2 \end{array})$ ?

Show answer & reasoning

$(4, 2)$ .

The input ${(0, 1)}^{T}$ selects the second column, so its image is ${(4, 2)}^{T}$ . Equivalently, the first column has weight 0 and the second column has weight 1.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Find the image of $(2, - 1)$ under $(\begin{array}{cccccccccc} 1 & 3 \\ 2 & 0 \end{array})$ .
2 marks
Write the matrix sending $e_{1}$ to $(2, 0)$ and $e_{2}$ to $(0, - 1)$ .
2 marks
Apply $(\begin{array}{cccccccccc} 0 & - 1 \\ 1 & 0 \end{array})$ to $(3, 2)$ .
2 marks
Apply $(\begin{array}{cccccccccc} 2 & 1 \\ 0 & 1 \end{array})$ to the vertices $(0, 0), (1, 0), (0, 2)$ .
2 marks
Find the matrix for $(x, y) \mapsto (x - 2 y, 3 x + y)$ .
2 marks
A map sends $e_{1}$ to $(a, b)$ and $e_{2}$ to $(c, d)$ . Find the image of $(3, - 2)$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the image of $(2, - 1)$ under $(\begin{array}{cccccccccc} 1 & 3 \\ 2 & 0 \end{array})$ .
2 marks
Multiply row by column to get $(2 - 3, 4 + 0)$ .
$(- 1, 4)$ .
Write the matrix sending $e_{1}$ to $(2, 0)$ and $e_{2}$ to $(0, - 1)$ .
2 marks
Use the images as columns.
$(\begin{array}{cccccccccc} 2 & 0 \\ 0 & - 1 \end{array})$ .
Apply $(\begin{array}{cccccccccc} 0 & - 1 \\ 1 & 0 \end{array})$ to $(3, 2)$ .
2 marks
The rule sends $(x, y)$ to $(- y, x)$ .
$(- 2, 3)$ .
Apply $(\begin{array}{cccccccccc} 2 & 1 \\ 0 & 1 \end{array})$ to the vertices $(0, 0), (1, 0), (0, 2)$ .
2 marks
Apply the same rule to every vertex.
$(0, 0), (2, 0), (2, 2)$ .
Find the matrix for $(x, y) \mapsto (x - 2 y, 3 x + y)$ .
2 marks
Coefficients of $x$ give column 1; coefficients of $y$ give column 2.
$(\begin{array}{cccccccccc} 1 & - 2 \\ 3 & 1 \end{array})$ .
A map sends $e_{1}$ to $(a, b)$ and $e_{2}$ to $(c, d)$ . Find the image of $(3, - 2)$ .
2 marks
Use three times the first image minus twice the second.
$(3 a - 2 c, 3 b - 2 d)$ .

Checking saved progress…

Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: matrices and transformations; original lessons follow the approved school-sized chapter structure.

Official curriculum source ↗

In this lesson

Understand the idea.

Images of the basis vectors

Map every vertex consistently

See the method in action.

Read the columns

Read the first unit-vector image.

Read the second unit-vector image.

Express the point using the unit vectors.

Apply the same weights to their images.

Construct a matrix

Put the given images into columns.

Multiply by the requested input.

Simplify the image coordinates.

Try it yourself.

Check your understanding

Keep practising