Lecture 13 · Week 4· 17:05

Linear transformations of Rⁿ

Linearity, standard matrix, rotations, shears, projections, reflections.

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In this lecture

Today: linear transformations
Transformations and linearity
First consequences
Linear or not?
The standard matrix
Finding a standard matrix
Rotations
Scalings and shears
Projections
Reflections
Summary

Materials

Try it

A matrix moves the whole grid

a top left0.00b top right0.00c bottom left0.00d bottom right0.00

Drag the tips of e₁ and e₂, use the sliders, or pick an example. The first column of A is where e₁ lands, the second is where e₂ lands; the shaded square shows how areas scale.

Check yourself

Try it before you look

Show that $T (x_{1}, x_{2}) = (x_{2}, x_{1} - 3 x_{2})$ is linear and find its standard matrix.
Find the standard matrix of the rotation through $18 0^{\circ}$ , and describe the map in words.
Show that $T (x_{1}, x_{2}) = (x_{1} x_{2}, 0)$ is not linear.
Find the standard matrix of the projection onto the second axis, and of the reflection across it.

Work it out on paper first. Then open the hint if you are stuck, and the answer to check.

1Hint

Everything you need is in Section 3.2 of the course book. Open it at page 9 of the chapter PDF; any theorem the question names is stated there, with an example worked the same way.

2Answer

$T (x) = A x$ with $A = [01 1 - 3]$ , so $T = T_{A}$ is linear (Proposition 3.2.2). Its columns are $T (e_{1}) = (0, 1)$ and $T (e_{2}) = (1, - 3)$ , so $A$ is the standard matrix.
$R_{18 0^{\circ}} = [- 1 0 0 - 1]$ , so $x \mapsto - x$ : a half turn, the point reflection through $0$ .
$T (2 (1, 1)) = T (2, 2) = (4, 0)$ , but $2 T (1, 1) = 2 (1, 0) = (2, 0)$ . The scaling rule fails, so $T$ is not linear.
Projection onto the second axis: $P = [0001]$ . Reflection across it: $Q = [- 1 0 01]$ .