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78 Cards in this Set

  • Front
  • Back
If a transformation is through an n x m matrix, what is the dimensional change?
From R^m to R^n
When is k |x|
a dilation and when is it a contraction?
Dilation - when K is greater than 1
Contraction - when K is between 0 and 1
How can you describe a projection in words?
proj(x) means turning x into something x(parallel) and x(perpendicular)

How do you calculate a projection onto a line spanned by a vector?
1. Normalize the vector
2. proj(x) = (x•u)u
How do you describe ref(x) algebraically?
ref(x) = 2proj(x) - x = 2(x•u)u - x
How do you describe a reflection in words?
Flipping a vector over something = x(parallel) - x(perpendicular)
What conditions have to be fulfilled in order for a matrix to be invertibel?
1. It must be square
2. rref(A) = 1
If T is transofmred from R^m to R^n, describe the image and the kernel.
im(T) is the subset of the codomain R^n
ker(T) is the domain of R^m
How do you describe the inverse of a 2x2 matrix, A = |a b|
|c d|
A^-1 = 1/(det(A)) |d -b|
|-c a|
If it is in the form y1 = ax1 + bx2 + cx3, what form is the inverse in?
Inverse is in the form, x1 = ay1 + by2 + cx3
What is hte kernel of an invertible nxn matrix?
ker = {0}
Is im(A) necessarily equal to im(rref(A))?
For an invertible nxn matrix, is im(A) = R^n?
im(A) = R^n
What is the rank?
Dimension of the image.
Is the image dependent on kernel?
What does it mean to be orthogonal?
Dot product is 0
What does orthonormal mean?
Unit vector are orthogonal to one another.
Are orthonormal vectors linearly independent?
What do orthnormal vectors form?
Basis of R^n.
How would you describe proj(x) in words?
An orthogonal projection amounts to porjectings it onto the x-axis, then onto the y-axis, then adding the resultant vectors.
How does you describe proj(x) algebraically?
(u1 • x1) u1 + ... + (uN • xN) uN
What is orthogonal complement?
The set of vectors in R^n that are orthogonal to all vectors in V.
What is the orthogonal complement the kernel of?
The orthogonal projection on to V.
What is the dimension of the subspace + the dimsneion of the kernel equal to?

For example, what is the kernel of a plane in R^3?
What is the kernel of a line in R^3?

Line: 1 + 2 = 3
Plane: 2 + 1 = 3

What is V ∩ V⊥?
What condition do 2 vectors x and y fit ≡ they are orthogonal complements?
||x + y||^2 = ||x||^2 + ||y||^2
Which is always greater than or equal to the other: x or proj(x)?
x by the Pythagorean Theorem
What is the angel between 2 vectors?
arc cos (x•y/(||x|| ||y||))
What is the correlation coefficient between two vectors?
r = cos(Θ) = (x•y/(||x|| ||y||)

What is another way to write v1 • v2 using Θ?
v1•v2 = ||v1|| ||v2|| cos(Θ)
If basis of V is v1 and v2, then how would you find the orthonormal basis?
1. u1 = normalized v1
2. u2 = u2 - (u1•v2)u1

(v3 would equal u3 - (u1•v3)u1 - (u2•v3)u2/ ||"|| )
Describe the relationship between basis and image with the kernel.
Basis is found in the kernel, so it involves the kernel.

Image does not involve the kernel.
What is an orthonormal transformation?
1. Preserves length
2. If put something with length one in, get length one out
3. All of its columns are linearly independent
4. All of its columns have length one.
For an nxn matrix, how can you compare the determinants of A and A^T?
det(A) = det(A^T)
What do the columns of an orthogonal matrix form?
An orthonormal basis of R^n
Is the product of 2 orthogonal matrices also orthogonal?
Is the inverse of an orthogonal matrix orthogonal?
Is matrix A is nxm, what is a matrix A^T?
What is a square matrix?
A = A^T
What is a skew-symmetric matrix?
-A = A^T
What is the general form a of skew matrix?
0 a b
-a 0 c
-b -c 0
If a matrix is orthogonal, what is A x A^T?
Identity Matrix
Is rank(A) = rank(A^T)?
Is Ker(A) = Ker (AA^-1)?
No, Ker(A) = Ker(AA^T)
What does it mean if ker(A) = 0?
AA^T is invertible
What are the least squares solutions of Ax=b?
x* = (A^TA)^-1 (A^T) (b)

(This is a matrix)
What type of entity is the least squares solution?
It is a matrix.
Describe the determinant in words?
The product of the values along the diagnoal.
What is the form of a discrete trajectory?
x(t+1) = Ax(t)
What is the form of the characteristics equation?
What is the characteristic equatin of an n x n matrix?
(-λ)^n + tr(A) (-λ)^(n-1) + ... + det(A) = 0
What is trace?
Sum of the diagonal entries.
What is algebraic multiplicity?
# of times a value is a root.
Do A and A^T have the same eigenvalues?
Do A and A^ T have the same characteristic polynomials?
What does it mean that 2 matrices are similar?
They have the same det and tr
How do you find closed formula, (like lake pollutant quesiton)?
1. x0 is the initial vector

2. x0 = c1v1 + c2v2 + ... + cnvn (Find eigenbasis to find c's)

3. x(t) = A^t x0 = c1 λ1^t v1 + ... + cn λn^t vt

4. Plug in values
How do you know if an nxn matrix is diagonizable?
If the dimensions of the iegnebasis add up to n
What does this mean?:

lim A^t
Diagonalize A
What is the relationsihp between A, S, and D?
A = (S^-1) (D) (S)
How can you find the closed formula solution?
x(t) = A^t x0

1. Use D = (S^-1) (A) (S) to find D
2. A^t = (S^-1) (D^t) (S)
3. Since x(t) = (A^t) (x0), x(t) = (S^-1) (D^t) (S) )x0)
When it says find closed formula, whta does it mean in other words?
Find all entries of A^t.
What is DeMoivre's formula?
(cosΘ + i sinΘ)^n = cos(nΘ) + i sin(nΘ)
If a 2 x 2 matrix has eigenvalues of a +ib and a - ib, what is S^-1 A S equal to?
The matrix:
a -b
b a
If z = a + bi, how do you find r and Θ?
r = root (a^2 + b^2)

Θ = arctan (b/a)
What do you do when you want to find some x(t + 1) = Ax(t) fraction as lim goes to infinity?
Find eigenbasis, assuming λ = 1, and normalize it.
When is A orthonogally diagonizable?
If it is symmetric, (A^T= A)
What are 2 ways to model evolution over time?
Discrete and continuous cases.
Describe a discrete case in words.
It's like bank accounts tht compound 7% interest each year.
Describe a discrete case algebraically.
X(t+1) = 1.07 x(t)
Describe a continuous case in words.
It's like bank accounts with 7% annual interest, compounded continuously.
Describe a continuous case algebrically.
dx/dt = 0.07 x ~ x = x0 e^(0.07t)
How do you find solutions to a discrete case, dx/dt = Ax, given A?
1. Find the eigenvectors of A and designate this as S.
2. Plug λs and eignebasises into the equation x(t) = c1 e^(λ1t) v1 + ... + cn e^(λnt) vn
- You can solve for c's using x0
-The dominant term si associated with the larger eigenvalue, because e is raised to a larger exponent. It, therefore, determines the behavior of the system in the distance future.
Describe the dominant term in a discrete case.
the dominant term is associated with the larger eigenvalue, because e is raised to a larger exponent. It, therefore, determine teh behavior of the system in the distant future.
Rewrite: e^(it)=cos(t) + i sint(t)
e^(p+iq)t = e^(pt) (cos(qt) + i sin(qt))
When is the continuous solution, dx/dt = Ax, a stable equilibrium solution?
When all real parts of the eigenvalues are negative.
What is the zero state of the discrete solution stable?
tr(A) is less than 0
det(A) is greater than 0
What do you do if you are asked to solve a discrete system, where A is a 2x2 matrix with complex eigenvalues?
1. Subtract positive λ from the diagonal
2. Find the kernel of this matrix
3. Seperate into a [real coefficients] + i [complex coefficients]
- Designate S = [real coefficients complex coefficients]
4. Plug into x(t) = e^(pt) • S • the matrix:
cos(qt) -sin(qt)
sin(qt) cos(qt)
• S^(-1) • x0