Linear Algebra Vocab

Linear Equation

Linear Equation

An equation of the form a1x1+a2x2+…+anxn=b

System of linear equarions

A collection of one or more linear equations involving the same variables

Solution

A list of numbers that makes each equation true when Si is subbed for xi

Solution set

The set of all possible solutions

Equivalent

Two linear systems have the same solution set

Coefficient matrix

The matrix containing only coefficient columns

Augmented matrix

The matrix containing coefficient and solution columns

Row Echelon Form

1. All non-zero rows above any zero rows

2. Each leading entry of a row is in a column to the right of the leading entry of the row above it

3. All entries in a column below a leading entry are zero

Reduced Row Echelon Form

Row Echelon form AND

4. The leading entry of each nonzero row is one

5. Each leading entry is the only non-zero entry in its column

Row Equivalent

When two matrices can be converted into the other by a series of elementary row operations

Pivot Position

A location in a matrix that corresponds to a leading entry in the reduced row echelon form

Pivot Column

A column of a matrix that contains a pivot

Equal (vectors)

Two vectors are equal if their corresponding entries are equal

Linear Combinations

Given vectors v1…vp and scalars c1…cp, the vector y is given by y=c1v1+…+cpvp (i.e v1…vp with weights c1…cp)

Span

The set of all linear combinations of v1, v2,…vp

Linear Function

1. T(u+v)=A(u+v)=Au+Av

2. T(cu)=A(cu)=cAu

Homogeneous Linear system

Ax=b is homogeneous if b=0 (Ax=0)

Linearly Independent

The vector equation for a set of vectors (x1v1+x2v2+…+xpvp=0) has ONLY the trivial solution

Linearly Dependent

There are scalars, not all zero, that the vector equations have non-trivial solutions (there are free variables)

Linear

1. T(u+v)=T(u)+T(v)

   1. T(cu)=cT(u)

Onto

T:Rn→Rm is onto IF each b in Rm is the image of at least one x in Rn such that T(x)=b (T is onto iff the columns of A span Rm)

One-to-One

T:Rn→Rm is one-to-one if each b in Rm is the image of AT MOST one x in Rn such that T(x)=b (T is one-to-one iff the columns of A are linearly independent)

Cramers Rule

Let A be an invertible nxn matrix. For every b in Rn, the unique solution x of Ax=b has entries Xi = (det(Ai(b)))/det(A)

Vector Space

A nonempty set V of objects on which are defined two operations called addition and scalar multiplication.

Subspace

A subspace of a vector space V is a subset, H of V, that has the following 3 properties

1. 0 is in H

2. H is closed under vector addition

3. H is closed under scalar multiplication

Column Space of A

ColA is the Span of the vectors in A

Kernel of T

KerT={x E V: T(x)=0}

Null Space of A

If A in an mxn matrix, the NulA={x E Rn: Ax=0}

Basis

Let H be a subset of vector space V. An indexed set B={b1,…, bp} of vectors in V is a basis for H is

1. B is linearly independent

2. SpanB=H

Dimension

The dimension of V (dimV) is the number of vectors in any basis for V

Rank of A

The dimension of the Column space of A (# of pivot columns)

Dimension of the Null of A

Number of free variables

Eigenvector

An eigenvector of an nxn matrix A is a non-zero vector x such that Ax=λx for some scalar λ

Eigenvalue

λ is an eigenvalue of A is there is a non-zero vector x in Rn such that Ax=λx

Probability Vector

A vector whose entries sum to 1

Stochastic Matrix

nxn matrix each of whose column vectors is a probability vector

Regular Stochastic Matrix

A stochastic Matrix each of whose entries is positive

A Markov Chain

A sequence of probability vectors such that xk+1=Pxk where P is a stochastic matrix

Steady-State Vector

Vector for a stochastic matrix P is a probability vector s such that Ps=s

Diagonalize

If A is an nxn matrix, we diagonalize A by finding a diagonal matrix D and an invertable matrix P such that A=PDP-1