How do you find coordinate vectors + what is their geometric meaning?
For some v = c1u1 + c2u2 + ... + ckuk 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.
Are orthogonal sets of vectors linearly indepedent?
Yes
What makes a linear transformation?
For some transformation T:
1) T(u + v) = T(u) + T(v)
2) T(cu) = cT(u)
How do you compute [T]C←B?
[[T(b1)C [T(b2)]C ... [T(bn)]C]
For two linear transformations T and S, which transformation acts first for [T ∘ S]?
S
No
How do you project a vector onto a subspace?
Project the vector onto each of the individual orthogonal basis vectors and adding them together
What should be the first thing that should be checked to see if a transformation is linear?
How do you compute [T]εm←εn?
[[T(e1) [T(e2)] ... [T(en)]] = [T]
What is a quick way to identify that a linear transformation is not invertible?
Difference in dimension when taking the transformation
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
How do you find an orthonormal basis?
T/F: Not every matrix transformation is a linear transformation.
How do you compute [T]εm←B?
[[T(b1) [T(b2)] ... [T(bn)]]
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])
T/F: (PC←B)-1 = PB←C
T
How do you check if a matrix is orthogonal?
For some orthogonal matrix A:
AAT = ATA = I; A-1 = AT
How is the standard matrix [T] built?
[T] = [T(e1) T(e2) ... T(en)]
How do you compute [T] given
[T]εm←B = [T]εm←εnPεn←B
Since [T] = [T]εm←εn
[T] = [T]εm←BPB←εn
[T] = [T]εm←B(Pεn←B)-1
[T] = [T(b1) ... T(bn)][b1 ... bn]-1
What does it mean if a function is injective?
For every distinct input there is a distinct output
What basis can be used as an intermediate when changing between two bases?
Standard basis
What is the determinant of an orthogonal matrix?
det(A) = ±1
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]]
If B is an orthonormal basis, how can the following be rewritten?
[S] = Pεn←B[S]B←BPB←εn
Since B is an orthonormal basis (and therefore orthogonal): PB←εn = (Pεn←B)-1 = (Pεn←B)T
[S] = Pεn←B[S]B←B(Pεn←B)T
When is a linear transformation one-to-one?
ker(T) = {0} or nullity(T) = 0