math 102 final review part 5!!!!!

Coordinate Vectors Relative to a Basis I

100

How do you find coordinate vectors + what is their geometric meaning?

For some v = c₁u₁ + c₂u₂ + ... + c_ku_k where the vectors are apart of an ordered basis, the scalars make up the coordinate vector for that basis

The scalars that make up the coordinate vector tells you how to get v by going c₁ units in u₁, etc.

200

Does using the standard basis output a new coordinate vector?

No

300

Given two bases B and C, how do you find the change of basis matrix from B to C?

Find the RREF of [C | B] and read off the right-hand side

400

T/F: (P_C←B)^-1 = P_B←C

T

500

What basis can be used as an intermediate when changing between two bases?

Standard basis

Coordinate Vectors Relative to a Basis II

100

Are orthogonal sets of vectors linearly indepedent?

Yes

200

How do you project a vector onto a subspace?

Project the vector onto each of the individual orthogonal basis vectors and adding them together

300

How do you find an orthonormal basis?

Graham Cracker Process

400

How do you check if a matrix is orthogonal?

For some orthogonal matrix A:

AA^T = A^TA = I; A^-1 = A^T

500

What is the determinant of an orthogonal matrix?

det(A) = ±1

Linear Transformations I

100

What makes a linear transformation?

For some transformation T:
1) T(u + v) = T(u) + T(v)
2) T(cu) = cT(u)

200

What should be the first thing that should be checked to see if a transformation is linear?

T(0) = 0

300

T/F: Not every matrix transformation is a linear transformation.

F

400

How is the standard matrix [T] built?

[T] = [T(e₁) T(e₂) ... T(e_n)]

500

What are the standard rotation matrices about:

1) the x-axis
2) the y-axis
3) the z-axis

1) [[1 0 0], [0 cosθ -sinθ], [0 sinθ cosθ]]
2) [[cosθ 0 sinθ], [0 1 0], [-sinθ 0 cosθ]]
3) [[cosθ -sinθ 0], [sinθ cosθ 0], [0 0 1]]

Linear Transformations II

100

How do you compute [T]_C←B?

[[T(b₁)_C [T(b₂)]_C ... [T(b_n)]_C]

200

How do you compute [T]_εm←εn?

[[T(e₁) [T(e₂)] ... [T(e_n)]] = [T]

300

How do you compute [T]_εm←B?

[[T(b₁) [T(b₂)] ... [T(b_n)]]

400

How do you compute [T] given

[T]_εm←B = [T]_εm←εnP_εn←B

Since [T] = [T]_εm←εn

[T] = [T]_εm←BP_B←εn
[T] = [T]_εm←B(P_εn←B)^-1
[T] = [T(b₁) ... T(b_n)][b₁ ... b_n]^-1

500

If B is an orthonormal basis, how can the following be rewritten?

[S] = P_εn←B[S]_B←BP_B←εn

Since B is an orthonormal basis (and therefore orthogonal): P_B←εn = (P_εn←B)^-1 = (P_εn←B)^T

[S] = P_εn←B[S]_B←B(P_εn←B)^T

Linear Transformations III

100

For two linear transformations T and S, which transformation acts first for [T ∘ S]?

S

200

What is a quick way to identify that a linear transformation is not invertible?

Difference in dimension when taking the transformation

300

Define:

1) Kernel
2) Image

1) The set of vectors that get sent to zero (similar to Null space) -> a subspace of domain -> ker(T) = Null([T])
2) The output(s) of the transformation (similar to codomain/range) -> a subspace of codomain -> im(T) = Col([T])

400

What does it mean if a function is injective?

For every distinct input there is a distinct output

500

When is a linear transformation one-to-one?

ker(T) = {0} or nullity(T) = 0