Linear Algebra

Front 1

How do you know if a matrix is singular?

Back 1

If it's Det = 0.

Front 2

What does having a singular matrix mean?

Back 2

That the matrix vectors are lin dependant.

Front 3

What is the det of a triangular matrix?

Back 3

The product of all its diagonal values.

Front 4

What is the det of the Id. matrix?

Back 4

1

Front 5

What is Det ( A -1 ) ?

Back 5

1/ Det (A)

Front 6

Det ( AB ) =

Back 6

Det ( BA )

Det (A) * Det (B)

Det (B) * Det (A)

Front 7

IF V is an n x n triangular matrix, then Det (V ^t) =

Back 7

Det (V)

Front 8

A - L*I is singular if...

Back 8

Det (A - L*I) = 0

Front 9

A scalar L is an eigenvalue of A if and only if ...

Back 9

A - L*I is singular

and

Det (A - L* I) = 0

Front 10

If A is an n x n matrix, then Det (A - L*I) is a ....

Back 10

ploynomial of degree n in t.

Front 11

What kind of vectors will span all of R^n?

Back 11

ANY number n vectors which are not scalar multiples of each other will span all of R^n.

Cool.

Front 12

How can you use scalars c1 through cn to produce formulas for the span of a set of vectors?

Back 12

Multiply each vector in the set by a unique scalar cn and concatenate the matrix with x, y, z, etc. then solve for each scalar to get formulas in x, y, z..

Front 13

What are the two conditions for a basis of a vector space?

Back 13

That they span that vector space.

That they are linearly independent.

Front 14

If a set of vectors spans S, then we automatically know that it is a ....

Back 14

subspace of S.

Front 15

What is the definition of a subspace?

Back 15

Any set of vectors which satisfy the two axioms of a vector space.

Closed under addition and under scalar multiplication.

Front 16

How do you find the eigenvalues of A^2?

Back 16

If Av = Lv

Then A^2v = A (Av) = A (Lv) = L (Av) = L (Lv) = L^2v.

Just square the eigenvalue.

Front 17

How do you find the eigenvalues of A ^ -1?

Back 17

If Av = Lv , then v = (Av) / L since L must be nonzero.

A ^-1 v = A^ -1 (Av / L) = A^ -1 A ( v / L) = v(1 / L)

Front 18

How do you find the eigenvalues of (A ^ -1 - I) ?

Back 18

By Av = Lv

(A ^ -1 - I)v = ( 1 / L - I)v

Front 19

What does it mean for Lamda to be an eigenvalue of A?

Back 19

It means that A - LI is singular.

When its determinant = 0

Front 20

How do you find the null space of a set of vectors and what is the definition?

Back 20

All vectors for which Ax = 0

Set the concatenated augmented matrix to RREF and then solve the system resulting from it.

Front 21

What is the definition for the range of a set of vectors?

Back 21

The set of all vectors y in R^m such that Ax = y is consistent.

Front 22

K is a nonzero scalar, and kA has an inverse, what is it?

Back 22

(1 / k) A ^ -1

Front 23

What is the inverse of A ^ t?

Back 23

( A ^ -1) ^t

Front 24

If A has an inverse, then what is A*A^ -1?

Back 24

A*A ^-1 = A^ -1 * A = I

A matrix multiplied either way by its inverse = the identity matrix.

Front 25

If the Det of a matrix = 0, what does this mean?

Back 25

That the matrix cannot have an inverse. It is singular. So, the vectors in the matrix are linearly dependent, there is no unique solution for this set of vectors.

Front 26

What is the formula for the inverse of a matrix if the determinant is not equal to 0?

Back 26

The matrix is nonsingular, the vectors are lin independent.

A^ -1 = 1/Det * [ d -b -c a ]

Front 27

What is the idea behind solving a system of equations using its matrix and its inverse?

Back 27

By Ax = b we have x = A^-1 b

Take the coefficient matrix, find its inverse using the identity matrix, then multiply the inverse matrix by the vector b on the RHS, this will give you your variables x1, x2..... xn. You can then verify that they are correct and satisfy the original equations.

Front 28

How can I get Q ^ -1 if Q = A^t * B^t?

Back 28

Q^ -1 = (A^ -1)^t * (B^-1)^t

Front 29

What is the formula for the angle between two vectors?

Back 29

u*v = [u][v]cos(theta)

Front 30

What the formula for the projection of vector b onto vector a?

Back 30

p = b*a / [a] * a / [a]

So, proj b onto a is just: b times a squared over length of a squared.

Front 31

What is the strategy for finding eigenvalues?

Back 31

Multiply your matrix by Lamda times the identity matrix, take the determinant of that, set it = 0, solve for all values of lamda.

Front 32

What is the strategy for finding eigenvectors?

Back 32

Take your eigenvalues one at a time, plug them into A-L*I, then solve that homogenous system.

Front 33

If you raise L to power of some integer k, how does that relate to your original matrix?

Back 33

Raise the original matrix to the same power to retain the relationship, so, if you raise a mtrix to a power, then you know that you can raise your L values to the same power and do not need to find them again.

Front 34

If the det of A is nonzero, then it is nonsingular and the vectors of A are lin independent. Then A has an inverse, assuming that A has eigenvalues L, how do they relate to A^ -1?

Back 34

1 / L is an eigenvalue of A^ -1

Front 35

For any scalar alpha added to lamda (L + a), how does that relate to the original matrix A?

Back 35

(L + a) = ( A + a*I)

This means that if you want to add values to either your original matrix or to the found lamdas, you can still preserve your relationship between the two.

Front 36

Do A and A^ t have the same eigenvalues?

Back 36

You betcha!!! :D

Front 37

If you happen to get a triangular matrix, what is the shortcut to finding your eigenvalues quickly?

Back 37

You eigenvalues will simply be the values of the diagonal entries.

Easy peasy.

Front 38

What makes a matrix defective?

Back 38

If the geometric multiplicity is less than the algebraic multiplicity.

If lamda has algebraic multiplicity 2 but there is only one vector in the eigenspace, then the matrix is defective.

Front 39

What is the formula that you want to use to diagonalize a matrix?

Back 39

You'll be wanting that the diagonal matrix D = S^ -1 * A * S

So then you get S from the eigenvectors of A. BAM!!! :D

Front 40

What is diagonal matrix D ^10 having to do with matrices A and S such that S can diagonalize A?

Back 40

D ^10 = S^ -1 * A^10 * S

Front 41

If you have D^ 10 = S^ -1 * A^10 * S.... how can you get what just A^10 = ...?

Back 41

A^ 10 = S* D^10 & S^-1


So, just flip everything backwards and you're good.

Front 42

What's the gameplan for diagonalizing a matrix?

Back 42

1) Find eigenvalues and form eigenvectors.

2) The eigenvectors give you the matrix you'll use to diagonalize the one that the eigenvalues came from.

3) Plug your matrices into the formula D = S^-1 * A * S

4) Ta-da!!! $$$$$$$ yeah.

Front 43

What is the definition of an orthogonal matrix?

Back 43

Q^t * Q = I

The transpose times the original matrix equals the identity.

Front 44

How do you find the least squares solution so an inconsistent system?

Back 44

Use the formula A^t*A*x = A^t*b

Multiply out, then you have two or three equations, solve for your x's all in terms of b, then your vector is b times whatever values it holds for the various x's.