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1

Linear Equation

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

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System of linear equarions

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

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Solution

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

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Solution set

The set of all possible solutions

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Equivalent

Two linear systems have the same solution set

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Coefficient matrix

The matrix containing only coefficient columns

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Augmented matrix

The matrix containing coefficient and solution columns

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Row Echelon Form

All non-zero rows above any zero rows

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

All entries in a column below a leading entry are zero

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Reduced Row Echelon Form

Row Echelon form AND

The leading entry of each nonzero row is one

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

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Row Equivalent

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

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Pivot Position

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

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Pivot Column

A column of a matrix that contains a pivot

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Equal (vectors)

Two vectors are equal if their corresponding entries are equal

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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)

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Span

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

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Linear Function

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

T(cu)=A(cu)=cAu

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Homogeneous Linear system

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

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Linearly Independent

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

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Linearly Dependent

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

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Linear

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

T(cu)=cT(u)

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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)

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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)

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Cramers Rule

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

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Vector Space

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

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Subspace

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

**0**is in HH is closed under vector addition

H is closed under scalar multiplication

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Column Space of A

ColA is the Span of the vectors in A

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Kernel of T

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

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Null Space of A

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

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Basis

Let H be a subset of vector space V. An indexed set B={**b**1,…, **b**p} of vectors in V is a basis for H is

B is linearly independent

SpanB=H

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Dimension

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

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Rank of A

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

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Dimension of the Null of A

Number of free variables

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Eigenvector

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

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Eigenvalue

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

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Probability Vector

A vector whose entries sum to 1

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Stochastic Matrix

nxn matrix each of whose column vectors is a probability vector

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Regular Stochastic Matrix

A stochastic Matrix each of whose entries is positive

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A Markov Chain

A sequence of probability vectors such that **x**k+1=P**x**k where P is a stochastic matrix

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Steady-State Vector

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

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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

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