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Tuesday
1. Matrices for linear transformations:
(a) Leftovers
(b) For each of the following matrices, describe the effects of its transformation:
5
12
−
3 −4
−1 16
3 0
−1 0
13
13
(d)
(e)
(a)
(b)
(c) 12
5
4 3
0 3
0 −1
2 − 13
13
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Related question: Which one of the preceding matrices represent isometries? In
order for a matrix to represent isometry, what must be true of its column vectors?
Why does that guarantee an isometry?
(c) Perform the following multiplication using the definition given yesterday then
interpret the process using transformations:
⎡ ⎤
1
1 2 3 ⎣ ⎦
2
3 2 1
3
(d) Find the entries of the following matrices:
2 × 2 matrix M for the reflection across the line y = x.
2 × 2 matrix N for the 90◦ counterclockwise rotation about the origin.
product MN; what transformation does this represent?
product NM; what transformation does this represent?
product MM; what transformation does this represent?
3 4
−5 5
(e) The matrix M = 4 3 defines an isometry of the xy- plane.
i.
ii.
iii.
iv.
v.
the
the
the
the
the
5
5
i. What special properties do the column vectors of this matrix have?
ii. Verify that the point (2, 4) remains stationary when M is applied to it. What
might M be?
iii. What is MM? What does this suggest about the geometric transformation
that M represents.
−0.6 0.8
0.8 0.6
(f) Matrices M =
and N =
represent reflections in the lines
0.8 0.6
0.6 −0.8
y = 2x and 3y = x. Verify that MN is not equal to NM, and explain why this
should have been expected. What transformations do the two products represent?
(g) Can you find a matrix for the reflection in y = ax?
(h) Give a general way of determining if the matrix is invertible.
inverse for an invertible matrix?5
⎡
1 0
⎣
(i) Consider the transformation represented by the matrix 0 1
0 0
effect on the points in the plane z = 1?
5
Can you find the
⎤
2
3⎦. What is its
1
Bonus investigation: Show that every invertible transformation is a product of reflections in lines through
0, and stretches in the x-direction.
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2. Directions
• Through the origin
• Linear independence
• Lines
1. Use vectors to describe the line through points (−1, 2) and (9, −2).
2. Use vectors to describe the line −3x + 4y = 12
3. Use vectors to describe the line −3x + 4y = 8
• Generalized directions
• Parallels
3. Invariants of linear transformations
4. Affine transformations
5. Vector Thales
6. Prove: Diagonals of parallelogram bisect each other
7. Centroid of a triangle:
−−→
(a) Draw a triangle ABC and let N be the midpoint of segment AC. Express BN in
−→
−→
terms of n = AB and m = AC.
−−→
(b) Let M be the midpoint of BC. Write AM in terms of m and n.
−→
−−→
−−→
(c) Express AB + 23 BN in terms of m and n. Express 23 AM in terms of m and n.
Hmmm....
−→ −−→
(d) Let G be the centroid of the triangle ABC. Simplify the vector sum GA + GB +
−→
GC.
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