Linear Algebra Vocab

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

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

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

The set of all possible solutions

Two linear systems have the same solution set

The matrix containing only coefficient columns

The matrix containing coefficient and solution columns

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

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

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

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

A column of a matrix that contains a pivot

Two vectors are equal if their corresponding entries are equal

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)

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

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

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

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

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

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

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

   1. T(cu)=cT(u)

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)

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)

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)

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

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

ColA is the Span of the vectors in A

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

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

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

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

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

Number of free variables

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

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

A vector whose entries sum to 1

nxn matrix each of whose column vectors is a probability vector

A stochastic Matrix each of whose entries is positive

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

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

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