Linear Algebra Test 1

Front 1

Gaussian Elimination

Back 1

When you solve a system like a normal person. The resulting matrix is in echelon form.

Front 2

echelon form

Back 2

A matrix has this form if...

1. Every leading term is in a column to the left of the leading term of the row below it.

2. Any zero rows are at the bottom of the matrix

Front 3

reduced row echelon form

Back 3

A matrix has this form if...

1. It is in echelon form

2. All pivot positions contain a 1

3. The only nonzero term in a pivot column is in the pivot position.

Front 4

Gauss-Jordan Elimination

Back 4

If you want to do extra work for no reason. The resulting matrix is in reduced row echelon form.

Front 5

pivot elements

Back 5

The circled elements

Front 6

trivial solution

Back 6

The solution where everything equals 0 and then everyone parties

Front 7

consistent system

Back 7

If the system has 1 solution or infinite solutions

Front 8

inconsistent system

Back 8

If the system has no solutions

Front 9

system of linear equations

Back 9

if you don't know what this is, you're doomed

Front 10

Matrix operations

Back 10

You are allowed to...

1. Interchange 2 equations

2. Multiply equations by nonzero constant

3. Add a multiple of one equation to another

I mean, duh...

Front 11

linearly independent

Back 11

If the only solution to the system is the trivial solution

Front 12

linearly dependent

Back 12

If there is a nontrivial solution

Front 13

homogeneous equation

Back 13

If the equation equals 0 (there aren't constants)

Front 14

homogeneous system

Back 14

If all the equations in the system equal 0

Front 15

row vector

Back 15

a vector that looks like that

Front 16

column vector

Back 16

a vector that looks like that

Front 17

algebraic properties of vectors

Back 17

literally learned these in every math/physics class since high school

Front 18

linear combination

Back 18

yeah man

Front 19

span

Back 19

The set of all possible linear combinations of x1u1 + x2u2 + . . . + xmum in R^n

written as: span{u1, u2, ... , um}

Front 20

How do you tell if a given vector is in the span of vectors?

Back 20

Look at the augmented matrix. The system has a solution iff the given vector is in the span.

Front 21

Theorems about span

Back 21

1. If u1, u2, ... un are vectors in R^n and b is in span{u1, ... un} then span{u1...um} = span{b, u1,...um}

2. If u1, ... um are vectors in R^n and m < n, then u1,...um do not span R^n

Front 22

nxm matrix

Back 22

Has n rows and m columns

Front 23

Does the matrix span R^3?

Back 23

No. Since row of 0's, there exists a vector not in the span of the column vectors, so the columns of the matrix do not span R^3

Front 24

Theorems about linear independence/dependence

Back 24

1. Suppose {0, u1, ... um} is a set of vectors in R^n. Then this set is linearly dependent.

2. Suppose {u1, u2, ... um} is a set of vectors in R^n. If n < m, then the set of vectors is linearly dependent.

3. Let {u1, ... um} be a set of vectors in R^n. The set is linearly dependent iff one of the vectors is in the span of the other vectors

Front 25

particular solution

Back 25

a fixed solution to the system

Front 26

homogeneous solution

Back 26

the solution to the homogeneous system

Front 27

Theorem about vectors...

Back 27

Let a1, a2, ... am and b be vectors in R^n. Then the following statements are equivalent.

1. the set {a1, ... am} is linearly independent

2. the vector equation x1a1 + ... + xmam =b has at most 1 solution.

3. The linear system corresponding to [a1, a2, ... am, b] has at most 1 solution

4. The equation Ax = b with A = [a1, a2, ... am] has at most 1 solution

Front 28

Big Theorem

Back 28

Let a be a set of vectors in R^n and let A = [a1, ... an] and let T: R^n -> R^n be given by T(x) = Ax Then the following are equivalent....

a.) a spans R^n

b.) a is linearly independent

c.) Ax = b has a unique solution for all b in R^n

d.) T is one-to-one

e.) T is onto

f.) matrix A is invertible

g.) Ker(T) = {0}

h.) A is a basis for R^n

Front 29

linear transformation

Back 29

A function T: R^m -> R^n is a linear transformation if for all vectors u and v in R^n and all scalars r,

(a) T(u + v) = T(u) + T(v)

(b) T(ru) = rT(u)

or combine (a) and (b) to get:

T(ru + sv) = rT(u) + sT(v)

Front 30

domain

Back 30

is R^m

Front 31

codomain

Back 31

is R^n

Front 32

image of u under T

Back 32

is T(u)

Front 33

range of T

Back 33

The set of all image vectors. Denoted range(T)

All y such that T(x) = y where T: R^m -> R^n

Front 34

linear transformation theorems

Back 34

1. Let A be an nxm matrix and define T(x) = Ax. Then T: R^m -> R^n is a linear transformation.

2. Let A = [a1 ... am] be an nxm matix and let T:R^m -> R^n with T(x) = Ax be a linear trans.

(a.) The vector w is in range(T) iff Ax = w is a consistent linear system

(b.) range(T) = span{a1, ..., am}

Front 35

one-to-one

Back 35

If T:R^m -> R^n is a linear transformation, T is one-to-one if for every vector w in R^n, there exists at most one vector u in R^n such that

T(u) = w.

Also, if T(u) = T(v) then u = v

Front 36

onto

Back 36

If T: R^m -> R^n is a linear transformation, T is onto if for every vector w in R^n there exists at least one vector u in R^m such that T(u) = w.

Front 37

Theorem about one-to-one

Back 37

Let A be an nxm matrix and define T: R^m -> R^n by T(x) = Ax. Then...

(a.) T is 1-1 iff the columns of A are linearly independent

(b.) If n < m then T is not one-to-one

Front 38

Theorem about onto

Back 38

Let A be an nxm matrix and define T: R^m -> R^n by T(x) = Ax. Then...

(a.) T is onto iff the columns of A span the codomain R^n

(b.) If n > m, then T is not onto

Front 39

How could a transformation be both one-to-one and onto?

Back 39

The number of columns and the number of rows in the matrix must equal each other. Square matrices for life.

Front 40

projection transformation

Back 40

Something that's like that

Front 41

matrix multiplication

Back 41

(mxn)(nxp) = mxp

Front 42

Samantha

Back 42

a genius

Front 43

Identity matrix

Back 43

durdurdur

Front 44

Matrix properties

Back 44

A(BC) = (AB)C

A(B + C) = AB + BC

(A + B)C = AC + BC

s(AB) = (sA)B or (A)(sB)

AI = A

IA = A

Front 45

Theorem with non-zero matrices

Back 45

Let A, B, and C be non-zero matrices.

(a.) It is possible that AB doesn't equal BA

(b.) AB = 0 doesn't imply A = 0 or B = 0

(c.) AC = BC doesn't imply that A = B or C = 0

Front 46

Theorem with matrix transforms

Back 46

A and B are both nxm matrices and C is an mxk matrix and s is a scalar

(a.) (A+B)^T = A^T + B^T

(b.) (sA)^T = s(A^T)

(c.) (AC)^T = (C^T)(A^T)

Front 47

transform of matrix

Back 47

make the rows the columns

Front 48

diagonal matrix

Back 48

diagonal with rest zeros

Front 49

upper triangular matrix

Back 49

numbers in upper right make a triangle

Front 50

lower triangular matrix

Back 50

numbers in lower left make a triangle

Front 51

invertible transformation

Back 51

If T: R^m -> R^n is one-to-one and onto

Front 52

inverse function for transformation

Back 52

yep

Front 53

inverse/linear transformation theorem

Back 53

Let T: R^m -> R^n be a linear transformation. Then...

(a.) T has an inverse only if m = n

(b.) If T is invertible, then T^-1 is also a linear transformation

Front 54

invertible matrix

Back 54

An nxm matrix A is invertible if there exists an nxm matrix B such that AB = In

Front 55

invertible matrix (obvious) theroem

Back 55

Suppose A is invertible with AB = In. Then BA = In and the matrix B is such that AB = BA = In and B is unique.

Front 56

inverse of matrix

Back 56

If the nxm matrix A is invertible, then A^-1 is called the inverse of A and is the unique matrix such that A(A^-1) = In and (A^-1)A = In

Front 57

nonsingular

Back 57

a matrix that is invertible

Front 58

singular

Back 58

a matrix that is not invertible

Front 59

Theorem about lots of invertible matrix stuff

Back 59

Let A and B be invertible nxn matrices and C and D be nxm matrices. Then...

(a.) AB is also invertible, (AB)^-1 = (B^-1)(A^-1)

(b.) A^-1 is invertible with (A^-1)^-1 = A

(c.) If AC = AD, then C = D

(d.) If AC = 0nxm then C = 0nxm

Front 60

How to find the inverse of a matrix

Back 60

There's some "shortcut" for a 2x2 matrix, but that's just another poop formula to memorize and no one wants that.

Front 61

Elementary Matrices

Back 61

E3 * E2 * E1 * A = u. u is A in REF. E's are identity matrices transformed by the operations used to turn A into u. (Example in notes from 2/12/16)

Front 62

subspace

Back 62

A subset S of R^n is a subspace if...

(a.) S contains 0 (the 0 vector)

(b.) If u and v are in S, then u + v is also in S

(c.) For r, a real number and u in S, ru is also in S

Front 63

subspace theorems

Back 63

1. Let S = span{u1, u2, ... um} be a subset of R^n Then S is a subspace of R^n.

2. If A is an nxm matrix, then the set of solutions to the homogeneous linear system Ax = 0 forms a subspace of R^m

Front 64

trivial subspace

Back 64

S = {0} is a subspace of R^n.

S = R^n is a subspace of R^n.

Those 2 are the trivial subspaces of R^n

Front 65

null space

Back 65

If A is an nxm matrix, then the set of solutions to Ax = 0 is called the null space of A and is denoted by null(A)

still don't really understand how you write this out... she's so dang confusing (and doesn't know what she's doing,) but whatever

Front 66

kernel of T

Back 66

The set of vectors x such that T(x) = 0.

denoted ker(T)

Front 67

kernel/range theorem

Back 67

Let T: R^m -> R^n be a linear transformation. Then... the kernel of T is a subspace of the domain (R^m) and the range of T is a subspace of the codomain (R^n).

Front 68

general theorem... (I don't know... It actually has a special name unlike all the other theorems in my notes...)

Back 68

Let T:R^m -> R^n be a linear transformation.

Then... T is 1-1 iff ker(T) = {0}

Front 69

basis

Back 69

A set B = {u1, ... um} is a basis for a subspace if

(a.) B spans S

(b.) B is linearly independent

Front 70

basis theorems

Back 70

Let B = {u1, ... um} be a basis for a subspace S. Then every vector s in S can be written as a linear combination s = s1u1 + ... + snun in exactly one way.

Front 71

subspace of rows theorem

Back 71

Let A and B be equivalent matrices (meaning one is transformed to the other by row operations.) Then the subspace spanned by the rows of A is the same as the subspace spanned by the rows of B.

Front 72

dimension theorem

Back 72

If S is a subspace of R^n, then every basis of S has the same number of vectors.

Front 73

dimension

Back 73

the dimension of S is the number of vectors in a any basis of S.

denoted dim(S)

Front 74

equivalent matrices

Back 74

2 matrices are equivalent if one is transformed to the other by row operations

Front 75

theorem about equivalent matrices

Back 75

Suppose U = [u1, ... um] and V = [v1, ... vm] are 2 equivalent matrices. Then any linear dependence that exists among vectors u1 ... um also exists among vectors v1, ..., vm.

yo, this is kinda common sense.

Front 76

theorem about forming basis for a subspace

Back 76

Let set U= {u1,...,um} be a set of vectors in a subspace S doesn't equal {0} or R^n.

(a.) If U is linearly independent, then either U is a basis for S or additional vectors can be added to U to form a basis

(b.) If U spans S, then either U is a basis for S or vectors can be removed from U to form a basis

Front 77

theorem about vectors forming basis that's pretty much the same thing as the other theorem about it

Back 77

Let U = {u1,...,un} be a set of vectors in subspace S of dim(S) = m. If U is either linear independent or spans S, then U is a basis for S

Front 78

subspaces/dimension theorem

Back 78

Suppose S1 and S2 are both subspaces of R^n and S1 is a subset of S2. Then...

dim(S1) <= dim(S2)

and dim(S1) = dim(S2) only if S1 = S2

Front 79

theorem about subspaces and dimension again

Back 79

Let U = {u1, ... um} be a set of vectors in a subspace S of dimension k.

(a.) If m < k, then U doesn't span S

(b.) If m > k, then U is linearly dependent

Front 80

standard basis

Back 80

{e1, e2,...,en} form a basis for R^n and are called the standard basis

Front 81

nullity

Back 81

the nullity of a matrix A is the dimension of the null space of A

denoted by nullity(A)

special case: If you get the 0 vector for the null space, then the nullity for the matrix is 0.