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Section3.3Matrix Equations

Objectives
  1. Understand the equivalence between a system of linear equations, an augmented matrix, a vector equation, and a matrix equation.
  2. Characterize the vectors b such that Ax = b is consistent, in terms of the span of the columns of A .
  3. Characterize matrices A such that Ax = b is consistent for all vectors b .
  4. Recipe: multiply a vector by a matrix (two ways).
  5. Picture: the set of all vectors b such that Ax = b is consistent.
  6. Vocabulary word: matrix equation.

Subsection3.3.1The Matrix Equation Ax = b .

In this section we introduce a very concise way of writing a system of linear equations: Ax = b . Here A is a matrix and x , b are vectors (generally of different sizes), so first we must explain how to multiply a matrix by a vector.

When we say A is an m × n matrix,” we mean that A has m rows and n columns.

Definition

Let A be an m × n matrix with columns v 1 , v 2 ,..., v n :

A = C ||| v 1 v 2 ··· v n ||| D

The product of A with a vector x in R n is the linear combination

Ax = C ||| v 1 v 2 ··· v n ||| DEIIG x 1 x 2 ... x n FJJH = x 1 v 1 + x 2 v 2 + ··· + x n v n .

This is a vector in R m .

In order for Ax to make sense, the number of entries of x has to be the same as the number of columns of A : we are using the entries of x as the coefficients of the columns of A in a linear combination. The resulting vector has the same number of entries as the number of rows of A , since each column of A has that number of entries.

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

The following properties are easy to verify.

Definition

A matrix equation is an equation of the form Ax = b , where A is an m × n matrix, b is a vector in R m , and x is a vector whose coefficients x 1 , x 2 ,..., x n are unknown.

Matrix Equations and Vector Equations

Let v 1 , v 2 ,..., v n and b be vectors in R m . Consider the vector equation

x 1 v 1 + x 2 v 2 + ··· + x n v n = b .

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

A = C ||| v 1 v 2 ··· v n ||| D and x = EIIG x 1 x 2 ... x n FJJH .

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

x 1 v 1 + x 2 v 2 + ··· + x n v n = b ,

where v 1 , v 2 ,..., v n are the columns of A , and x 1 , x 2 ,..., x n are the entries of x .

Four Ways of Writing a Linear System

We now have four equivalent ways of writing (and thinking about) a system of linear equations:

  1. As a system of equations:
    M 2 x 1 + 3 x 2 2 x 3 = 7 x 1 x 2 3 x 3 = 5
  2. As an augmented matrix:
    K 23 2 71 1 3 5 L
  3. As a vector equation (x 1 v 1 + x 2 v 2 + ··· + x n v n = b ):
    x 1 K 21 L + x 2 K 3 1 L + x 3 K 2 3 L = K 75 L
  4. As a matrix equation (Ax = b ):
    K 23 21 1 3 LC x 1 x 2 x 3 D = K 75 L .

In particular, all four have the same solution set.

We will move back and forth freely between the four ways of writing a linear system, over and over again, for the rest of the book.

Another Way to Compute Ax

The above definition is a useful way of defining the product of a matrix with a vector when it comes to understanding the relationship between matrix equations and vector equations. Here we give a definition that is better-adapted to computations by hand.

Definition

A row vector is a matrix with one row. The product of a row vector of length n and a (column) vector of length n is

A a 1 a 2 ··· a n BEIIG x 1 x 2 ... x n FJJH = a 1 x 1 + a 2 x 2 + ··· + a n x n .

This is a scalar.

Recipe: The row-column rule for matrix-vector multiplication

If A is an m × n matrix with rows r 1 , r 2 ,..., r m , and x is a vector in R n , then

Ax = EIIG r 1 r 2 ... r m FJJH x = EIIG r 1 xr 2 x ... r m x FJJH .

Subsection3.3.2Spans and Consistency

Let A be a matrix with columns v 1 , v 2 ,..., v n :

A = C ||| v 1 v 2 ··· v n ||| D .

Then

Ax = b hasasolution ⇐⇒ thereexist x 1 , x 2 ,..., x n suchthat A EIIG x 1 x 2 ... x n FJJH = b ⇐⇒ thereexist x 1 , x 2 ,..., x n suchthat x 1 v 1 + x 2 v 2 + ··· + x n v n = b ⇐⇒ b isalinearcombinationof v 1 , v 2 ,..., v n ⇐⇒ b isinthespanofthecolumnsof A .
Spans and Consistency

The matrix equation Ax = b has a solution if and only if b is in the span of the columns of A .

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

When Solutions Always Exist

Building on this note, we have the following criterion for when Ax = b is consistent for every choice of b .

Proof

Recall that equivalent means that, for any given matrix A , either all of the conditions of the above theorem are true, or they are all false.

Be careful when reading the statement of the above theorem. The first two conditions look very much like this note, but they are logically quite different because of the quantifier “for all b ”.