Matrices acting on position vectors | Year 11

01 / LEARN

The idea, step by step

Images of the basis vectors

The first column of A is the image of $e_{1} = {(1, 0)}^{T}$ , and its second is the image of $e_{2} = {(0, 1)}^{T}$ . Since ${(x, y)}^{T} = x e_{1} + y e_{2}$ , its image is x times the first column plus y times the second.

This is a constructive method: knowing the images of the two unit vectors determines the entire linear map. Put those images into columns, not rows.

Map every vertex consistently

Apply the same matrix to every vertex of a polygon. Connect corresponding images in the original order. A linear map sends a line to a line or collapses it to a point.

The grid illustrates the original triangle and its image under (x,y) to (x+y,y). Compute vertex coordinates to check a sketch; visual appearance alone is not proof.

A common trap

Putting the images of the unit vectors into rows.

Instead: With column-vector convention, those images are the columns.

02 / WORKED EXAMPLES

See the thinking

Example 1

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 $(2, 1)$ .

The two columns are ${(2, 1)}^{T}$ and ${(- 1, 3)}^{T}$ .
Compute $2 (2, 1) + (- 1, 3) = (3, 5)$ .

Unit-vector images

(2, 1)

and

(- 1, 3)

; point image

(3, 5)

Example 2

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 $(1, 2)$ .

Use these images as columns.
Multiply the resulting matrix by ${(1, 2)}^{T}$ .

A = (\begin{array}{cccccccccc} 0 & - 3 \\ 2 & 0 \end{array})

, image

(- 6, 2)

Check your understanding

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

Show answer

$(4, 2)$ .

Read the second column.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF 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.
$(3 a - 2 c, 3 b - 2 d)$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

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

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