Prelim 2 (2.3-5.5)

Question 1

Invertible Linear Transformation Theorem

Answer 1

T is invertible if A is invertible

Question 2

3 conditions for a subspace

Answer 2

1. Includes the zero vector
2. Closed under vector addition
3. Closed under scalar multiplication

Question 3

Theorem for Nul A

Answer 3

Nul A is a subspace of R^n (# of columns)

Question 4

Nul A

Answer 4

set of all solutions to Ax=0

Question 5

Theorem of Col A

Answer 5

Col A is a subspace of R^m (# of rows)

Question 6

Col A

Answer 6

set of all linear combinations of the columns of A

Question 7

to check if u is in Nul A

Answer 7

Au must equal 0

Question 8

to check if v is in Col A

Answer 8

Ax=v must be consistent

Question 9

kernel / null space

Answer 9

set of all u in vector space V such that T(u)=0 (equal the zero vector in vector space W)

Question 10

range

Answer 10

set of all vectors in W (T(x)) for some vector x in V

Question 11

Theorem for linearly dependency

Answer 11

a set {v1,…,vp} is linearly dependent if some vj (j > 1) is a linear combination of the preceding vectors {v1…v(j-1)}

Question 12

Theorem for basis of Col A

Answer 12

the pivot columns of A form the basis for Col A

- row reduce to echelon form and match the pivot columns to those in original matrix A

Question 13

Spanning Set Theorem

Answer 13

Let S be a set of vectors {v1…vp} and H be Span{v1…vp}

1. If one vector in S that is a linear combination of the others is removed, the set still spans H.
2. If H=/={0}, some subset of S forms basis for H.

Question 14

The Uniqueness Representation Theorem

Answer 14

Let B={b1…bn} be a basis for a vector space V. For each x in V, there exists a unique set of scalars c1…cn such that c1b1+…+cnbn=x

Question 15

coordinate mapping theorem

Answer 15

the coordinate mapping x->[x]b is a one to one linear transformation from Rn to Rn

Question 16

Change of Basis Theorem

Answer 16

Let B={b1…bn} and C={c1…cn} be basis for vector space V. There exists a unique n X n matrix such that [x]c=P(C

Question 17

Determinant and Invertible Matrices

Answer 17

if detA = 0, then A is not invertible (one of the entires on the main diagonal is 0 so A can’t be row equivalent to identity matrix)

Question 18

triangular matrix

Answer 18

• detA is the product of the entires on the main diagonal

- eigenvalues are on the main diagonal

Question 19

determinant of transpose of A

Answer 19

= determinant of A

Question 20

Area with Transformation

Answer 20

Let T be a linear transformation determined by a 2x2 matrix A. If S is a parallelogram in R2, then {area of T(S)} = (area of S) * |det A|

Question 21

Volume with Transformation

Answer 21

Let T be a linear transformation determined by a 3x3 matrix. If S is a parallelepiped in R3, then {volume of T(S)} = (volume of S) * |det A|

Question 22

eigenvector

Answer 22

a NONZERO VECTOR x such that Ax=λx

Question 23

Eigenvectors & linear dependency

Answer 23

If v1…vr are eigenvectors that correspond to distinct eigenvalues, then the set {v1…vr} is linearly independent.

Question 24

The Rank Theorem

Answer 24

rank + dim Nul A = n

Question 25

rank

Answer 25

dimension of the column space of A

Question 26

Row Space (Row A)

Answer 26

the set of all linear combinations of the rows of A

• reduce to echelon form and take the nonzero rows
• Row A = rank (transpose of A)
• if A and B are row equivalent, they have the same row space

Question 27

characteristic equation

Answer 27

det(A-λI)=0

• forms the characteristic polynomial
• has n roots, counting complex roots and multiplicities

Question 28

similar matrices

Answer 28

A is similar to B if there is an invertible matrix such that A = PB(P-1)

Question 29

similar matrices and eigenvalues

Answer 29

If A is similar to B, then they share the same characteristic polynomial and therefore the same eigenvalues with the same multiplicity.

Question 30

even if two matrices have the same eigenvalues

Answer 30

they are not necessarily similar

Question 31

Diagonalization Theorem

Answer 31

n x n matrix A is diagonalizable if it has n linearly independent eigenvectors.
For A=PD(P-1), the columns of P are n linearly independent eigenvectors of A. The diagonal entries of D are the corresponding eigenvalues to the eigenvectors in P.

Question 32

matrix A is said to be diagonalizable if

Answer 32

it is similar to diagonal matrix D

Question 33

Steps to diagonalize

Answer 33

1. Find the eigenvalues.
2. Plug in the eigenvalues and solve (A-λI)x=0. Get a good x (integers) and those are the eigenvectors.
3. Check eigenvectors are linearly independent. (there must be n of them)
4. The eigenvectors form the columns of P.
5. Construct D by having the diagonal be the eigenvalues corresponding to the columns of P.

Question 34

stochastic matrix

Answer 34

matrix whose columns are possibility vectors (all entries are positive)

Question 35

possibility vector

Answer 35

a vector whose entries are positive and add up to 1

Question 36

finding steady state vector for P

Answer 36

1. Solve (A-I)x=0.
2. Choose a simple basis (want integers)
3. To turn into possibility vector, add up the entries and divide the vector by that sum.

Question 37

a stochastic matrix is regular if

Answer 37

some matrix power P^k contains only positive entries

Question 38

Casorati matrix

Answer 38

Uk Vk W
Uk+1 Vk+1 Wk+1
Uk+2 Vk+2 Wk+2

Question 39

prove a solution form a basis (signals)

Answer 39

1. not multiplies of each other, linearly independent
2. (Casorati Matrix)(c1, c2, c3)=(0,0,0)
   plug in solutions for U, V, W and show that the casorati matrix is invertible after setting k=0 (columns are linearly independent)