Matrices as Transformations

The columns are where the basis vectors land.

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

NameDate

Figure — use these to answer the problems

The orange arrow is where the first basis vector lands, the purple where the second lands. Change a column and watch the whole grid follow.

a, first column top1

b, second column top1

c, first column bottom0

d, second column bottom1

v, first entry2

v, second entry1

Figure 1

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

The same ideas with more to keep track of.

3 problems

The identity matrix. Where does ⟨3, 4⟩ land? Enter the first entry.
Answer
A matrix with both columns ⟨1, 1⟩. Does it flatten the plane to a line? 1 yes, 0 no.
Answer
How many vectors decide a transformation of the plane completely?
Answer

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.
Write each item under the heading it belongs to: The origin stays at the origin · Grid lines stay straight and parallel · Lengths of vectors · Angles between vectors
Always preserved
May change
A matrix with columns ⟨2, 0⟩ and ⟨0, 2⟩. By what factor does it stretch every vector?
Answer
The Landing
A matrix has first column ⟨4, 3⟩. Where does ⟨1, 0⟩ land? Enter the second entry.
Answer
The Origin
Where does the origin land under any linear transformation? Enter the first entry.
Answer