Linear Algebra

Question 1

Equation to find eigenvalues

Answer 1

det(A-lambda*I)=0

Question 2

How to find eigenvectors

Answer 2

Solve for the vector (v) that satisfies the equation (A-lambdaI)v = 0.

Question 3

What does a determinant of zero indicate?

Answer 3

That the matrix does not have linear independence. At least one of the rows is a multiple of another row in the same matrix.

Question 4

When forming the elimination matrix, how is e_ij formed?

Answer 4

eij = -Aij/Ajj

Question 5

While forming a 3x3 elimination matrix, after obtaining E32, E21, and E32 separately, what is the shortcut to form E32E321?

Answer 5

a=E21
b=E31
c=E32

E32E321 will have “a” in the 2,1 spot, “c” in the 3,2 spot, and the 3,1 spot will be (a x c) + b

Question 6

Matrix elimination equation for a 3x3 matrix

Answer 6

E32E321A = E32E321b

NOTE: The key here is that E32 CANNOT be obtained without first finding Abar = E321*A. Once you get E32, you can form the E matrix above and multiply it by the original A.

Question 7

When forming an elimination matrix of a matrix A, what should be done with matrix A once a new column in the elimination matrix has been completed?

Answer 7

Multiply what you have so far in the elimination matrix by Matrix A. You CANNOT begin to form a new column of E until this step is complete, or the answer will be WRONG.

Question 8

What is the elimination matrix of A equivalent to?

Answer 8

A^-1

Question 9

When forming the total elimination matrix E from matrix A, in what order should the columns of E be formed.

Answer 9

Start by forming the lower triangle. Begin with E21 and work to the bottom. Then multiply E and A and move to the next column. Next form the upper triangular by beginning with E12. Multiply E12 and the latest form of A together and then move on to the next column. Finally, finish by forming Ef to reduce all diagonal values of A to one.

Question 10

Equation for the complete 3x3 elimination matrix of A

Answer 10

Ef E123 E12 E32 E321 A

Question 11

How do you find the inverse of the elimination matrix E?

Answer 11

E^-1 is made up of the inverse of each of the sub elimination matrices, all multiplied together To invert a sub elimination matrix, simply swap the sign corresponding to the appropriate element.

Question 12

PA = LU

Answer 12

1) Apply the P matrix at the beginning of the process to get pivots in the correct spots.
2) U is simply the upper triangular matrix formed by using the lower elimination matrix on A.
3) Form L by taking the inverse of E. If E is Ef E123 E12 E32 E321, then L would be E321^-1 E32^-1 E12^-1 E123^-1.

Change the E on the inverted terms to an L, e.g., E321^-1 = L321.

Finally, tack on an Lf term at the beginning of L. Lf is the diagonal of A.

The final result looks like this: A = LPU or A = L321 L32 L12 L123 Lf P U

Question 13

What does the space of R^n consist of?

Answer 13

All column vectors v with n components

Question 14

What is a subspace?

Answer 14

A set of vectors (ALWAYS including zero) that satisfies two requirements. If v and w are vectors in the subspace and c is any scalar, then:

1) v + w is in the subspace
2) cv is in the subspace

Basically, all linear combinations of vectors in the subspace with still be in the subspace.

Question 15

How to find the span of a matrix A?

Answer 15

To find the span of b, break the columns of A up into vectors and multiply each one by x1, x2, x3, etc. Then, pick two sets of values for each of those variables. This will form two vectors. Then find a plane that each of those vectors lies in. Cross the vectors to get the normal vector to the plane. 
Use a(x-xo) + b(y-yo) + c(z-zo) = 0 to find the equation of the plane/subspace.

Question 16

Define column space

Answer 16

The column space includes all linear combinations of the columns. The combinations are all possible vectors Ax.

Question 17

Can Ax=b be solved if b is not in the column space of A?

Answer 17

No

Question 18

Define N(A), the nullspace of A

Answer 18

N(A) consists of all solutions to Ax=0. The vectors x are in R^n

Question 19

What are free variables

Answer 19

Free variables are the variables that will be multiplied to the free or dependent columns when trying to find N(A).

Question 20

What are free columns

Answer 20

Columns that depend on all of the independent columns before them. They come after columns that have pivot values of 1.

Question 21

What are the steps to find N(A)?

Answer 21

1) Reduce A to row echelon form
2) Locate the independent and the free columns, as well as the free variables
3) Form the S vectors that will make up the null space. There will be the same amount of S vectors as free columns. S vectors are determine by picking numbers for the free variables and then solving for the values of the independent variables.

Question 22

Full column rank

Answer 22

The columns of A are independent when the rank is r=n

Question 23

Is a set of vectors dependent or independent when n > m?

Answer 23

Dependent

Question 24

How to find the rank of a matrix

Answer 24

The rank is equal to the number of independent columns. This is also called the “true column space.” It is also the number of pivots

Question 25

When are the columns of A linearly independent?

Answer 25

When the only solution to Ax = 0 is x=0. (Vectors cannot simply be combined to give the solution x=0)

Question 26

Define row space

Answer 26

The row space of a matrix is the subspace of R^n spanned by the rows. It is also the column space of A^T

Question 27

What is a basis

Answer 27

A basis for a vector space is a sequence of vectors that:

1) are linearly independent
2) span the space

Question 28

What is the dimension of C(A) and C(A^T)?

Answer 28

r (rank of the matrix)

Question 29

Dimension of N(A)

Answer 29

n-r

Question 30

Dimension of N(A^T)

Answer 30

m-r

Question 31

Define C(A)

Answer 31

The column space of A is the subspace of R^m spanned by the columns

Question 32

N(A) is a subspace of…

Answer 32

R^n

Question 33

N(A^T) is a subspace of…

Answer 33

R^m