Matrices as Transformations

The columns are where the basis vectors land.

12 problems · about 23 minutes · N-VM.C.11, G-CO.A.2

What this lesson teaches

The student interprets a matrix as a transformation of the plane and reads its columns.

A matrix is a transformation that moves every point of the plane.
Its columns are exactly where the two basis vectors land, and everything else follows.
Grid lines stay straight, parallel and evenly spaced, and the origin never moves.

Straightforward practice. Get the method working first.

5 problems

The first column of a matrix is ⟨0, 6⟩. Where does ⟨1, 0⟩ land? Enter the second entry.
Answer 6
Why 6.
What do the columns of a matrix tell you?
Answer Where each basis vector lands after the transformation.
Why The images of the basis vectors.
The first column of a matrix is ⟨2, 5⟩. Where does ⟨1, 0⟩ land? Enter the first entry.
Answer 2
Why Read the first column.
The second column is ⟨0, 7⟩. Where does ⟨0, 1⟩ land? Enter the second entry.
Answer 7
Why Read the second column.
Does a linear transformation ever bend a straight line? 1 yes, 0 no.
Answer 0
Why Grid lines stay straight.

The same ideas with more to keep track of.

3 problems

The identity matrix. Where does ⟨3, 4⟩ land? Enter the first entry.
Answer 3
Why Nothing moves.
A matrix with both columns ⟨1, 1⟩. Does it flatten the plane to a line? 1 yes, 0 no.
Answer 1
Why Both basis vectors land in the same place.
How many vectors decide a transformation of the plane completely?
Answer 2
Why One per dimension.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each property by whether a linear transformation preserves it.
Answer Always preserved: The origin stays at the origin, Grid lines stay straight and parallel · May change: Lengths of vectors, Angles between vectors
Why Stretching changes size without bending anything.
A matrix with columns ⟨2, 0⟩ and ⟨0, 2⟩. By what factor does it stretch every vector?
Answer 2
Why Both directions double.
The Landing: A matrix has first column ⟨4, 3⟩. Where does ⟨1, 0⟩ land? Enter the second entry.
Answer 3
Why 3.
The Origin: Where does the origin land under any linear transformation? Enter the first entry.
Answer 0
Why 0.