A directed line segment that corresponds to a displacement from one point A to another point B

Vector

Initial point of vector

Tail

2 vectors that are scalar multiples of each other

Parallel vectors

A vector v is a linear combination of vectors v1, v2,…,vk if there are scalars c1,c2,…ck such that v=c1v1+c2v2+…+ckvk. The scalars c1,c2,…,ck are called the coefficients of the linear combination

Linear combination

Aka dot product

Scalar product

Length. llvll = √v.v

Norm of a vector

vector of length 1

Unit vector

finding a unit vector in same direction

normalizing

d(u,v)=llu-vll (can switch u and v) – u=(a,b) and v=(a1, b1): find the norm ‖u-v‖=√((a-a_1 )^2+〖(b-b_1)〗^2 )

Formula to find distance between vectors u and v

cos(theta) = (u.v)/(llullllvll)

Formula for angle between 2 vectors

Two vectors u and v are orthogonal if u.v=0

Orthogonal vectors

(u.v.)/(u.u) * u

Formula for projection of v onto u

Vector n that is orthogonal to any vector x that is parallel to the line

Normal vector to line

n.x = n.p (p is a specific point)

Normal form of line

ax+by=c

General form of equation of line

x=p+td (p is a specific point on line, d is direction vector)

vector equation of a line

|ax0+by0-c|/√(a^2+b^2 )

Formula for distance from point to line

|ax0+by0+cz0-d|/√(a^2+b^2+c^2 )

Formula for distance from point to plane

axb=(a2b3-a3b2, a3b1-a1b3, a1b2-b1a2)

Cross product formula

commutative, associative with addition and multiplication, distributive (can distribute scalar over vector sum or vector over scalar sum)

Algebraic properties of vectors in Rn

u.v=v.u

u.(v+w)=u.v + u.w

(cu).v =c(u.v)

u.u≥0 and u.u=0 iff u=0

u.(v+w)=u.v + u.w

(cu).v =c(u.v)

u.u≥0 and u.u=0 iff u=0

Properties of vector dot products

Only when v=0

When does llvll=0?

lcl llvll

llcvll =

lu.vl≤ llull llvll

Cauchy-Schwarz inequality

Opposite

If the angle between u and v is obtuse, then proju(v) will be in the same/opposite direction from u?

negative scalar multiple

If the angle between u and v is obtuse, proju(v) will be a ___ of u

direction vector

Parallel planes have the same __

u

Proju(v) is a scalar multiple of vector __

u and v are orthogonal

llu+vll = llu-vll iff

proju(v)

proju(proju(v)) =

no, infinite

Are vector and parametric forms of equation of line unique?

2-D

A plane is an x-D object

If (GCF of a,m) / b doesn’t have remainder, then # solutions if GCF (if remainder then no solutions)

How to determine how many solutions ax=b (mod m) has

Can do scalar addition with because scalar

What to remember about dot product (u.v)

sin(theta) = (u.v)/llullllvll

Formula to find angle between plane and line

An equation that can be written in form a1x1+a2x2+…+anxn where a1,a2,…,an,b are constants

Linear equation

In each nonzero, first nonzero entry is in column to left of any leading entries below it. All entries in a column below a leading entry (not necess. 1) are 0.

Row echelon form

Can do to matrices without changing solutions: 1. interchange 2 rows 2. multiply a row by a constant 3. add a multiple of one row to another row

Elementary row operations

to get ref

Gaussian elimination

If can perform row operations to make one look like other. Doesn’t mean they’re equivalent matrices

Row equivalent

If same rref

Easy way to determine if 2 matrices are row equivalent

The rank of a matrix is the number of non-zero rows in its row echelon form OR the # of leading variables OR (# variables)-(# free variables)

Rank – 3 definitions

Same

Rank of matrix vs. rank of its transpose

max: 3, min:0

What are max and min ranks of 5×3 matrix?

rank(A)

The row vectors of mxn matrix A are linearly dependent iff rank(A) ? n

In system of n variables, number of free variables = n-rank

Rank Theorem

All constant terms are 0

Homogenous system

line

If 1 direction vector, then equation of a:

plane

If 2 direction vectors, then equation of a:

Put into augmented matrix and consistent (meaning no 0 on left and nonzero on right). Then must make sure to plug into all equations, could be no solution…

How to tell if something is a linear combination?

If there’s at leas tone solution

Consistent

If s={v1,v2,…,vk} is a set of vectors in Rn, then the set of all linear combinations of v1,v2,…,vk is called the span of v1,v2,…,vk and is written (v1,v2,..,vk) or span(s)

Span

Means you can get to any point in R2 using linear combinations of u and v. Make x(u) + y(V)=[a,b] then create augmented matrix and rref.

To show that span(s)=R^2

At least 3 vectors

How many vectors do you need for a set that spans R3

A set of vectors v1,v2,…,vk is linearly dependent if there are scalars c1,c2,…,ck at least one of which is not zero, such that c1v1+c2v2+…+ckvk=0

Linear dependence

If have any zero rows, then linearly dependent

Relationship between zero rows and linear dependence

create a zero row

The rows of a matrix will be linearly dependent if elementary row operations can be used to:

Independent

If x=y=z=0, are vectors linearly dependent/independent?

Independent

Is the empty set linearly dependent/independent?

m>n

Any set of m vectors in Rn is linearly independent if:

1. Find scalars such that c1v1+…+ckvk=0 (one constant has to be non-zero)

2. Show that at least one of the vectors can be expressed as a linear combo of the others

3. Let v1,…,vn be column vectors in Rn and let A be the matrix with these vectors as columns. Show that augmented matrix [Al0] has nontrviail solution

2. Show that at least one of the vectors can be expressed as a linear combo of the others

3. Let v1,…,vn be column vectors in Rn and let A be the matrix with these vectors as columns. Show that augmented matrix [Al0] has nontrviail solution

3 ways to prove linear dependence

1. Show that c1v1+…+ckvk=0 has only trivial solution

2. Show [Al0] has only the trivial solution

2. Show [Al0] has only the trivial solution

2 ways to prove linear independence

matrix with 0s everywhere but diagonal, can also have 0s in diagonal

Diagonal matrix

special case of diagonal matrix, all entries in diagonal are same number

scalar matrix

exactly the same incl size and entries

equal matrices

when A=A^T

symmetric matrix

the transpose of an m x n matrix A is the n xm matrix A^T obtained by interchanging the rows and columns of A

transpose

A^r+s

(A^r)(A^s)=

A^rs

(A^r)^s=

All entries below main diagonal are 0

Upper triangular matrix

commutative, associative, distributivity over matrix and scalar addition

Properties of Addition and scalar multiplication

Ax=0

Homogeneous linear system

associative, distributes over addition, can commute the scalar – NOT commutative

Properties of matrix multiplication

k(A^T)

(kA)^T=

A^T + B^T

(A+B)^T=

(B^T)(A^T)

(AB)^T=

(A^T)^r

(A^r)^T =

If A is an nxn matrix, an inverse of A is an nxn matrix A’ with the property that AA’=I and A’A=I where I=In is the nxn identity matrix. If such an A’ exists, then A is called invertible

Inverse of a matrix

unique

If A is an invertible matrix, then its inverse is:

x=A^-1 * b for any b in Rn

If A is an invertible nxn matrix, then the system of linear equations given by Ax=b has the unique solution:

(1/ad-bc)[d -b/-c a]

Formula for inverse of 2×2 matrix

(1/c)A^-1

(cA)^-1=

B^-1 * A^-1

(AB)^-1=

(A^-1)^T

(A^T)^-1=

(A^-1)^n

(A^n)^-1 =

no because not commutative

Can you distribute for (A^-1B^3)^2 ?

can be made by performing one elementary row operation to identity matrix

Elementary matrices

invertible

Each elementary matrix is:

A^-1

If a sequence of elementary row operations reduces A to I then the same sequence of row operations transform I into:

A subspace of Rn is any collection S of vectors in Rn such that: 1. the zero vector is in s. 2. if u and v are in s, then u+v is in S. 3. If u is in S and c is a scalar, then cu is in S.

Subspace

one of the 3 stipulations has to fail

How to prove that something isn’t a subspace?

a point (origin), a line through origin, R2 (the whole thing)

What are possible subspaces in R2?

The rowspace of A is the subspace row(A) of Rn spanned by the rows of A

Row space

not the row space but DO change column space

Do elementary rows change the row space? column space?

The column space of A is the subspace col(A) of Rm spanned by columns of A

column space

Must be equal

Relationship between # of vectors in row and column spaces

place in augmented matrix with A. If consistent, means can find linear combos so vector is in col(A)

To determine if a vector is in col(A)

augment with A and see if consistent (note: can take transpose to make more familiar)

To determine if vector is in row(A)

Let A be an mxn matrix. The nullspace of A is the subspace of Rn consisting of solutions of the homogenous linear system Ax=0. It is denoted null(A)

Nullspace

yes, even if just rivial solution

Does every set of vectors have a nullspace? explain

A. There is no solution

B. There is a unique solution

C. There are infinitely many solutions

B. There is a unique solution

C. There are infinitely many solutions

For any system of linear equations Ax=b, what are the 3 options for results?

A basis for a subspace S of Rn is a set of vectors in S that: 1. spans S and 2. is linearly independent

Basis

always be the same

The number of vectors for a given subspace will

vectors that are scalar multiples/extraneous

Basis gets rid of:

rref matrix. Use nonzero row vectors of R (with leading 1s) to form basis for row(A)

How to find row space basis

rref matrix. Use column vectors of A corresponding to leading 1s to form basis for col(A)

How to find column space basis

rref matrix and augment with 0. Create x=,y=, z= etc. equations, all in terms of 1 free variable (must have as many equations as have variables!) – don’t forget z=z is an equation

How to find null space basis

free variable

Null space gets 1 vector for every:

row space vectors

Null space vectors are orthogonal to:

#columns in original matrix

(# null space vectors)+(#row space vectors OR #column space vectors)=

vectors

If S is a subspace of Rn, any 2 bases for S have the same # of:

If S is a subspace of Rn, then the number of vectors in a basis for S is called the dimension of S, denoted dimS

Dimension

zero vector

What is always a subspace of Rn?

Linearly dependent, so can’t have a basis

What 2 things are true of any set containing zero vector?

dimension

Row and column spaces of matrix have the same __

rank(A)

rank(A^T)=

The nullity of a matrix A is the dimension of its null space. In other words, dimension of solution of Ax=0, which is same as # free variables in the solution

Nullity

If A is an mxn matrix, then rank(A)+nullity(A)=n

The Rank Theorem

A transformation T:Rn–>Rm is called a linear transformation if: 1. t(u+v)=t(u)+t(v) for all u and v in Rn. 2. t(cv)=ct(v) for all u in Rn and c in R

Linear transformation

linear transformation

A vector multiplied by a matrix is a:

let u=[x1,y1] and v=[x2,y2] and show that 1.t(u+v) and t(u)+t(V) are equal, and 2. t(cv) and c(tv) are equal

how to show something is a linear ttransformation:

find one counterexample

how to prove that something’s not a linear transformation

reflection, rotation, dilation

Give 3 examples of linear transformations

translation because origin doesn’t get mapped to itself

give one example of not linear transformation, and why

take identity matrix and divide into 2 columns, treat each as point and see where would be mapped to

How to find matrix for linear transformation

[costheta -sintheta/sintheta costheta]

linear transformation matrix for rotation

Let A be an nxn matrix. A scalar λ is called an eigenvalue of A if there is a nonzero vector x such that Ax=λx

Eigenvalue

Let A be an nxn matrix. A vector x is called an eigenvector of A corresponding to λ if there is a nonzero vector x such that Ax=λx

Eigenvector

eigenvector and has same eigenvalue

Any scalar multiple of an eigenvector is also an ___ and has the same ___

2

Max # of possible eigenvectors for 2×2 matrix

Let A be an nxn matrix and let λ be an eigenvalue of A. The collection of eigenvectors corresponding to λ, together with the zero vector, is called the eigenspace of λ and is denoted by Eλ

Eigenspace

Multiply v by A to get [xa, xb] matrix where x is eigenvalue

How to show a vector is an eigenvector of A

(A-Iλ)x=0 then rref and create variable equations to find eigenvector

How to show that λ is an eigenvalue of A and find one eigenvector corr to eigenvalue

det(A-λI) to ind char polynomial, then set equal to 0 to find eigenvalues. (A-λI) for different eigenvalues to find eigenvectors/spaces

Find all eigenvalues and eigenvectors of A

ad-bc

Determinant of 2×2 matrix

scalar

Determinant is a __

the product of the entries on its main diagonal

The determinant of a triangular matrix is:

If A has a zero row or column, then detA=

-detA

If B is obtained by interchanging 2 rows or columns of A, then detB=

If A has 2 identical rows or columns, then detA=

kdetA

If B is obtained by multiplying a row or column of A by a constant k, then detB=

detA

If B is obtained by adding a multiple k of one row or column of A to another, then detB=

(k^n)det(A)

det(kA)=

det(AB)

If A and B are nxn matrices, then det(A)*det(B)=

1/(det(A))

If A is invertible, then det(A^-1)=

det(A)

For any square matrix A, detA^T=

find determinant in terms of k to find char polynomial. k can’t equal any values for which char polynomial would equal 0

For what values of k (components within marix) is A invertible? – how to solve

det(A-λI)

Characteristic polynomial of A

det(A-λI)=0

Characteristic equation of A

multiplicity of a root of characteristic polynomial

Algebraic multiplicity

number of vectors in a basis for an eigensapce

geometric multiplicity

entries on its main diagonal

Eigenvalues for a triangular matrix are:

0 isn’t an eigenvalue of A

A square matrix is invertible iff

A. A is invertible

B. Ax=b has a unique solution for every b in Rn

C. Ax=0 has only the trivial solution

D. The rref of A is In

E. A is a product of elementary matrices

F. rank(A)=n

G. nullity(A)=0

H. The column vectors of A are linearly independent

I. The column vectors of A span Rn

J. The column vectors of A form a basis for Rn

K. The row vectors of A are linearly independent

L. The row vectors of A span Rn

M. The row vectors of A form a basis for Rn

N. detA≠0

O. 0 is not an eigenvalue of A

B. Ax=b has a unique solution for every b in Rn

C. Ax=0 has only the trivial solution

D. The rref of A is In

E. A is a product of elementary matrices

F. rank(A)=n

G. nullity(A)=0

H. The column vectors of A are linearly independent

I. The column vectors of A span Rn

J. The column vectors of A form a basis for Rn

K. The row vectors of A are linearly independent

L. The row vectors of A span Rn

M. The row vectors of A form a basis for Rn

N. detA≠0

O. 0 is not an eigenvalue of A

Fundamental Theorem of Invertible Matrices

linearly independent

If A is an nxn matrix and λ1,λ2,…,λm are distinct eigenvalues of A with corresponding eigenvectors v1,v2,..,vm then v1,.v2,…,vm are:

Let A and B be nxn matrices. A is similar to B if there is an invertible matrix P such that (P^-1)AP=B

Similar matrices

AP=PB, don’t have to find P^-1 for similarity problems

What’s an equivalent equation to (P^-1)AP=B that can be helpful

A

If A~B then B~

A~C

If A~B and B~C then:

detB

If A~B, then detA=

B is invertible

If A~B, then A is invertible iff

rank, char polynomial, eigenvalues

If A~B, then A and B have the same: (3 answers)

B^m for all integers m≥0

If A~B, then A^m ~

B^m for all integers

If A~B, then if A is invertible, A^m ~

Showing two matrices aren’t similar bc can’t be if properties fail

What are properties of similar matrices most helpful for?

First detA≠detB. If not, then A and B don’t have same char polynomial

What are 2 ways to immediately tell matrices aren’t similar

An nxn matrix is diagonalizable if there is a diagonal matrix D such that A is similar to D – that is, if there’s an invertible nxn matrix P such that (P^-1)AP=D

Diagonalizable

matrix where columns are eigenvectors

What is P in (P^-1)AP=D diagonalization equation?

matrix where diagonal entries are eigenvalues

What is D in (P^-1)AP=D diagonalization equation?

n

If A is an nxn matrix, A is diagonalizable iff A has __ linearly independent eigenvectors.

no

If 3×3 matrix and 2 eigenvectors, is A diagonalizable?

diagonalizable

If A is an nxn matrix with n distinct eigenvalues, then A is:

geometric mult less than or equal to algebraic mult

If A is an nxn matrix, what is relationship between geometric and algebraic multiplicity of each eigenvalue?

Algebraic multiplicity of each eigenvalue equals its geometric multilicty

What is true of algebraic and geometric multiplicities of diagonalizable matrices?

columns

Not diagonalizable when number of eigenspaces doesn’t correspond with number of:

[a^n 0/0 b^n]

How to raise diagonal matrix to a power [a 0/0 b]^n=

λ^3=λ

λ^3-λ=0, then solve for λ

λ^3-λ=0, then solve for λ

What are possible eigenvalues of A if A^3=A?

A set of vectors {v1,v2,…,vk} in Rn is called an orthogonal set if all distinct pairs of vectors are orthogonal

Orthogonal set

linearly independent

If {v1,v2,…,vk} is an orthogonal set of nonzero vectors in Rn, then these vectors are:

An orthogonal basis for a subspace W of Rn is a basis of W that is an orthogonal set

Orthogonal basis

Must show that every pair of vectors from this set is orthogonal, or that v1.v2=0, v2.v3=0, v3.v1=0

How to show that a set of vectors is an orthogonal set given v1,v2,v3

(w.vi)/(vi.vi) for i=1,…,k

Let {v1,v2,…,vk} be an orthogonal basis for a subspace W of Rn and let w be any vectors in W. Then the unique scalars c1,…,ck such that w=c1v1+…+ckvk are given by ci=

A set of vectors in Rn is an orthonormal set if it is an orthogonal set of unit vectors

Orthonormal set

An orthonormal basis for a subspace W of Rn is a basis of W that is an orthonormal set

Orthonormal basis

An nxn matrix Q whose columns form an orthonormal set is called an orthogonal matrix

Orthogonal matrix

Q^T

A square matrix Q is orthogonal iff Q^-1=

an orthonormal set

If Q is an orthogonal matrix, then its rows form:

is orthogonal

If Q is orthogonal, then Q^-1

+/- 1

If Q is orthogonal, then detQ=

=1

If Q is orthogonal, then if λ is an eigenvalue of Q, then lλl:

Q1Q2

If Q1 and Q2 are orthogonal nxn matrices, then so is:

If v is orthogonal to every vector in W

When can you say a vector v is orthogonal to a subpsace W?

The set of all vectors that are orthogonal to W is called the orthogonal complement of W, denoted w(perp)

Orthogonal complement

a subspace of Rn

If W is a subspace of Rn, then W(perp) is:

W

(Wperp)perp =

nullspace of A

Orhotongal complement of rowspace of A is:

the nullspace of A^T

Orthogonal complement of column space of A is:

rowspace, column space, nullspace, nullspace of A^T

what are the 4 fundamental subspaces of A?

Let w be a subspace of Rn and let {u1,u2,…,uk} be an orthogonal basis for W. For any vector v in Rn, the orthogonal projection of v onto w is defined as projw(V) = ((u1.v)/(u1.u1)u1) + ((u2.v)/(u2.u2)u2)+…+((uk.v)/(uk.uk))uk

Orthogonal projection of v onto w

perpw(v) = v-projw(v)

Component of v orthogonal to w is the vector:

v-projw(v)

perpw(v) =

w+w(perp) = projw(v) + perpw(v)

Orthogonal Decomposition Theorem: v=

v1=x1

v2=x2-projv1(x2)

v3=x3-projv1(x3) – projv2(x3)

v2=x2-projv1(x2)

v3=x3-projv1(x3) – projv2(x3)

Gram-Schmidt Algorithm

Make any basis (with 3 linearly independent vectors). Then make perpendicular using Gram-Schmidt

Method for finding orthogonal basis for R3 that contains a given vector

A square matrix A is orthogonally diagonalizable if there exists an orthogonal matrix Q and a diagonal matrix D such that (Q^T)AQ=D

Orthogonally diagonalizable

If matrix is symmetric, it’s orthogonally diagonalizable

Relationship between symmetric and orthogonally diagonalizable matrix:

(Q^-1)AQ=D

What’s an equivalent statement for (Q^T)AQ=D?

v is a vector space (and its elements are called vectors) if:

1. closed for addition

2. commutativity works for addition

3. associative for addition

4. u+0=u

5. u+(-u)=0

6. closed for scalar multiplication

7. associative for scalar multiplication

8. distributive over scalar multiplication

9. distributive over scalar addition

10. identity (1u=u)

1. closed for addition

2. commutativity works for addition

3. associative for addition

4. u+0=u

5. u+(-u)=0

6. closed for scalar multiplication

7. associative for scalar multiplication

8. distributive over scalar multiplication

9. distributive over scalar addition

10. identity (1u=u)

Vector space (what 10 things must be true)

6-10

When looking for axiom that makes vector space fail and addition is same as usual, which ones to focus on?

1. if u and v are in w, then u+v in w

2. if u in w and c is scalar, then cu is in w

2. if u in w and c is scalar, then cu is in w

w is a subspace of vectorspace v iff what 2 things are true: