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 Product

Let 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)=cAuDefinition

A 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 Equations

Let 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:

As a system of 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 Ax

The 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.

Definition

A 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 multiplication

If 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.

Theorem

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