## Front Matter

## 1Systems of linear Equations: Algebra

## 2Systems of direct Equations: Geometry

## 3Linear Transformations and also Matrix Algebra

## 4Determinants

## 5Eigenvalues and Eigenvectors

## 6Orthogonality

## Back Matter

Section2.3Matrix Equations¶ permalinkObjectivesUnderstand the equivalence in between a mechanism of linear equations, one augmented matrix, a vector equation, and also a matrix equation.Characterize the vectors b such that Ax=b is consistent, in terms of the expectancy of the columns the A.Characterize matrices A such that Ax=b is continuous for all vectors b.

You are watching: Use the definition of ax to write the matrix equation as a vector equation.

*Recipe:*multiply a vector by a matrix (two ways).

*Picture:*the set of every vectors b such that Ax=b is consistent.

*Vocabulary word:*

*matrix equation*.Subsection2.3.1The matrix Equation Ax=b.

In this ar we present a really concise method of writing a system of linear equations: Ax=b. Here A is a matrix and x,b are vectors (generally of various sizes), so an initial we must define how to main point a procession by a vector.

When us say “A is one m×n matrix,” we median that A has actually m rows and n columns.

Remark

In this book, we perform *not* reserve the letters m and also n for the numbers of rows and columns of a matrix. If we write “A is one n×m matrix”, climate n is the variety of rows that A and m is the variety of columns.

Definition

Let A it is in an m×n matrix v columns v1,v2,...,vn:

A=C|||v1v2···vn|||D

The *product* the A v a vector x in Rn is the linear combination

Ax=C|||v1v2···vn|||DEIIGx1x2...xnFJJH=x1v1+x2v2+···+xnvn.

This is a vector in Rm.

Example

In order for Ax to make sense, the variety of entries the x needs to be the very same as the number of columns of A: we space using the entries the x together the coefficients the the columns the A in a linear combination. The resulting vector has actually the same number of entries together the number of *rows* of A, due to the fact that each pillar of A has that number of entries.

If A is one m×n matrix (m rows, n columns), climate Ax makes sense when x has n entries. The product Ax has m entries.

Properties that the Matrix-Vector ProductLet A be an m×n matrix, permit u,v be vectors in Rn, and also let c be a scalar. Then:

A(u+v)=Au+AvA(cu)=cAuDefinitionA *matrix equation* is one equation the the type Ax=b, wherein A is an m×n matrix, b is a vector in Rm, and x is a vector whose coefficients x1,x2,...,xn space unknown.

In this publication we will examine two safety questions about a procession equation Ax=b:

Given a specific selection of b, what are all of the options to Ax=b?What are all of the choices of b so that Ax=b is consistent?The first question is more like the questions you can be provided to indigenous your previously courses in algebra; you have a many practice addressing equations choose x2−1=0 for x. The second question is maybe a new concept because that you. The location theorem in Section 2.9, which is the culmination that this chapter, tells united state that the two inquiries are intimately related.

Matrix Equations and also Vector EquationsLet v1,v2,...,vn and b be vectors in Rm. Think about the vector equation

x1v1+x2v2+···+xnvn=b.

This is equivalent to the matrix equation Ax=b, where

A=C|||v1v2···vn|||Dandx=EIIGx1x2...xnFJJH.

Conversely, if A is any kind of m×n matrix, then Ax=b is identical to the vector equation

x1v1+x2v2+···+xnvn=b,

where v1,v2,...,vn space the columns of A, and also x1,x2,...,xn are the entries of x.

Example

Four ways of writing a direct System

We now have *four* equivalent ways of writing (and thinking about) a device of direct equations:

M2x1+3x2−2x3=7x1−x2−3x3=5

As one augmented matrix:

K23−2

71−1−35L

As a vector equation (x1v1+x2v2+···+xnvn=b):

x1K21L+x2K3−1L+x3K−2−3L=K75L

As a matrix equation (Ax=b):

K23−21−1−3LCx1x2x3D=K75L.

In particular, *all 4 have the same solution set*.

We will certainly move ago and soon freely between the four ways of composing a linear system, over and also over again, because that the remainder of the book.

Another method to Compute AxThe above an interpretation is a useful method of defining the product the a matrix with a vector as soon as it concerns understanding the relationship between matrix equations and vector equations. Here we provide a meaning that is better-adapted to computations by hand.

DefinitionA *row vector* is a matrix through one row. The *product* the a heat vector of length n and a (column) vector of length n is

Aa1a2···anBEIIGx1x2...xnFJJH=a1x1+a2x2+···+anxn.

This is a scalar.

Recipe: The row-column ascendancy for matrix-vector multiplicationIf A is an m×n matrix with rows r1,r2,...,rm, and x is a vector in Rn, then

Ax=EIIG—r1——r2—...—rm—FJJHx=EIIGr1xr2x...rmxFJJH.

Example

Subsection2.3.2Spans and also Consistency

Let A it is in a matrix v columns v1,v2,...,vn:

A=C|||v1v2···vn|||D.

Then

Ax=bhasasolution⇐⇒thereexistx1,x2,...,xnsuchthatAEIIGx1x2...xnFJJH=b⇐⇒thereexistx1,x2,...,xnsuchthatx1v1+x2v2+···+xnvn=b⇐⇒bisalinearcombinationofv1,v2,...,vn⇐⇒bisinthespanofthecolumnsofA.

Spans and Consistency

The procession equation Ax=b has actually a equipment if and also only if b is in the expectancy of the columns of A.

This gives an equivalence between an *algebraic* statement (Ax=b is consistent), and a *geometric* explain (b is in the expectancy of the columns that A).

Example(An Inconsistent System)

Example(A constant System)

When Solutions always Exist

Building ~ above this note, we have actually the adhering to criterion for when Ax=b is regular for *every* selection of b.

Let A it is in an m×n (non-augmented) matrix. The following are equivalent:

Ax=b has a solution for all b in Rm.The expectancy of the columns that A is all of Rm.Proof

The equivalence the 1 and 2 is developed by this keep in mind as used to every b in Rm.

Now we show that 1 and also 3 space equivalent. (Since we understand 1 and also 2 space equivalent, this indicates 2 and 3 are tantamount as well.) If A has a pivot in every row, climate its lessened row echelon type looks like this:

C10A0A01A0A0001AD,

and because of this AAbB reduces to this:

C10A0A

A01A0AA0001AAD.

There is no b that makes it inconsistent, so over there is constantly a solution. Whereas if A walk not have actually a pivot in every row, then its lessened row echelon form looks choose this:

C10A0A01A0A00000D,

which can give rise come an inconsistent mechanism after augmenting through b:

C10A0A

001A0A00000016D.

Recall that *equivalent* way that, for any given procession A, either *all* of the problems of the over theorem are true, or they are all false.

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Be careful when reading the declare of the above theorem. The very first two conditions look really much prefer this note, but they room logically quite different because of the quantifier “*for all* b”.

Interactive: The criteria the the theorem are satisfied

Interactive: The critera the the theorem are not satisfied

Remark