5.3 - Operators in R2 and R3

Question 1

Recall: What is a a vector space? What is Rn.

Answer 1

A vector space, V, is a set of vectors with two operation defined: Addition and Scalar Multiplication, satisfying the following axioms.
Of addition (v and w are vectors):
1. v and w in V, v + w in V
2, v + w = w + v
3. (v + w) + z = v + (w + z)
4. v + 0 = v
5. v + -v = 0
Of scalar multiplication (a and b are scalars):
1. If a is a real number and v is in V, av is in V
2. a(v + w) = av + aw
3. (a + b)v = av + bv
4. a(bv) = (a*b)v
5. 1v = v
Rn is a vector space containing all vectors comprised of real numbers with n components.

Question 2

Recall: What is a subset? Subspace?

Answer 2

Subset (or set I think): Any collection of vectors
Subspace: Collection of vectors with special restrictions
1. It contains the 0-vec
2. Adding any 2 vectors results in a vector also in the subspace
3. Scalar multiplication of a vector in the subspace results in a vector also in the subspace
4.

Question 3

Recall: What is a basis for a subspace?

Answer 3

Set of linearly independent vectors that span the subspace.

Question 4

Recall: What is a span of a set of vectors? Define linearly independency vs dependency of two or more vectors? What does a span of two linearly independent vectors entail?

Answer 4

The span of a set of vectors in the the collection of all possible linear combinations of that set of vectors. If a vector is said to be in the span of a set of vectors, in can be written as a linear combo of those vectors.
Linear independency: Vectors cannot be written as scalar multiples of each other
Linear Dependency: Vectors can be written as linear multiples of each other
If a span of two linearly dependent vectors exists, one is said to be redundant and it can be removed as it can span a set containing only the other vector.

Question 5

Def: What is the rotational matrix in R2.

Answer 5

Rtheta = (cosO -sinO; sinO cosO)

Question 6

Def: What is the Standard Matrix?

Answer 6

A standard matrix is like a transformation matrix. when applied to a vector, it transforms it.

Question 7

Def: What are the standard matrices for reflection in the y and x axis and on the origin in R2.

Answer 7

x: (1 0; 0 -1)
y: (-1 0; 0 1)
origin or y=x : (-1 0 : 0 -1)

Question 8

Def: What are the standard matrices for projection onto x and y axis in R2?

Answer 8

x: (1 0; 0 0)
y: (0 0; 0 1)

Question 9

Recall: What is the formula for projection of b onto a.

Answer 9

proj of b onto a = (b.a/a.a)*a

Question 10

Recall: What is a symmetric matrix

Answer 10

A matrix that is equivalent to it’s transpose.

Question 11

Def: What is the projection matrix?

Answer 11

a = (a1, a2), a vector.
proj of x onto a = ((a1^2/|a|^2) (a1a2/|a|^2) ; (a1a2/|a|^2) (a2^2/|a|^2)

Question 12

Def: What is the standard matrix of a projection onto any line? How can we use the projection matrix to determine it?

Answer 12

L is any line.
a is a vector.
x is a vector.
proj of x onto L = proj of x onto a if a//L.

If L is a line, then L is y = kx, k is any scalar number not zero. In vector form, L is (x, y), but writing y in terms of x, it is (x, kx). As a linear combination, x(1, k) (can also be done in terms of y).

Therefore a = x(1, k). We can choose any x value (usually 1). Once chosen, we can simplify a = (x, kx) to a = (c, d). The standard matrix of a projection onto the line a = (c, d) using the projection matrix is:
((c^2/|a|^2) (cd/|a|^2) ; (cd/|a|^2) (d^2/|a|^2)

Question 13

Theory: What is the rank of a standard projection matrix.

Answer 13

1. It is a line, so only one leading entry is possible.

Question 14

Def: Standard Matrix (SM) over y = x.

Answer 14

0 1
1 0

Question 15

Theory: How do projections of lines onto planes work.

Answer 15

Only one coordinate of the lines (not in the plane) is 0

Question 16

Def: SM of projection onto:
1. XY plane
2. XZ Plane
3. ZY Plane

Answer 16

1 0 0
0 1 0
0 0 0

1 0 0
0 0 0
0 0 1

0 0 0
0 1 0
0 0 1

Question 17

Theory: Explain rotations of points in 3 dimensions.

Answer 17

Imagine plane perpendicular to axis of rotation (if z was the axis, plane would be parallel to XY plane). The point exists on the plane. The plane is rotated CCW, so the coordinate on the axis of rotation (z in this case) is unchanged.

Question 18

Def: Rotation matrix on z and z axis.

Answer 18

1 0 0
0 cost -sint
0 sint cost

cost -sint 0
sint cost 0
0 0 1

Question 19

Def: Rotation in y axis. Why does it not contain the rotation matrix?

Answer 19

Using positive x and z axis, the plane between them would be XZ. Rotating the plane on Y, from X to Z, is CW, which is a negative angle, as opposed to Z to X, which is CCW. Therefore, in the SM, angle is negative.
cos(-t) = cost
sin(-t) = -sin(t)

cost 0 sint
0 1 0
-sint 0 cost

Question 20

Theory: How to find linear operator involving two transformations?

Answer 20

1. Obtain SM of both transfomations
2. Multiply both SM’s to get SM of the linear operator
3. Create transformation using SM

Question 21

Theory: Reflection in 3 dimensions over a plane for XY plane and how?

Answer 21

Switch the sign of the vector coordinate the same as the coordinate not on the plane.
1 0 0
0 1 0
0 0 -1