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Section3.2Vector Equations and Spans

Objectives
  1. Understand the equivalence between a system of linear equations and a vector equation.
  2. Learn the definition of Span { x 1 , x 2 ,..., x p } , and how to draw pictures of spans.
  3. Recipe: solve a vector equation using augmented matrices / decide if a vector is in a span.
  4. Pictures: an inconsistent system of equations, a consistent system of equations, spans in R 2 and R 3 .
  5. Vocabulary word: vector equation.
  6. Essential vocabulary word: span.

Subsection3.2.1Vector Equations

An equation involving vectors with n coordinates is the same as n equations involving only numbers. For example, the equation

x C 126 D + y C 1 2 1 D = C 8163 D (3.2.1)

simplifies to

C x 2 x 6 x D + C y 2 y y D = C 8163 D or C x y 2 x 2 y 6 x y D = C 8163 D .

For two vectors to be equal, all of their coordinates must be equal, so this is just the system of linear equations

E x y = 82 x 2 y = 166 x y = 3. (3.2.2)
Definition

A vector equation is an equation involving a linear combination of vectors with possibly unknown coefficients.

Figure3A picture of the vector equation (3.2.1). Try to solve the equation geometrically by moving the sliders.
A Picture of a Consistent System

We saw in the above example that the system of equations (3.2.2) is consistent. Equivalently, this means that the vector equation (3.2.1) has a solution. Therefore, the figure above is a picture of a consistent system of equations. Compare this figure below.

In order to actually solve the vector equation

x C 126 D + y C 1 2 1 D = C 8163 D ,

one has to solve the system of linear equations

E x y = 82 x 2 y = 166 x y = 3.

This means forming the augmented matrix

C 1 1 82 2 166 1 3 D

and row reducing. Note that the columns of the augmented matrix are the vectors from the original vector equation, so it is not actually necessary to write the system of equations: one can go directly from the vector equation to the augmented matrix by “smooshing the vectors together”.

Recipe: Solving a vector equation

In general, the vector equation

x 1 v 1 + x 2 v 2 + ··· + x p v p = b

where v 1 , v 2 ,..., v p , b are vectors in R n and x 1 , x 2 ,..., x p are unknown scalars, has the same solution set as the linear system with augmented matrix

C ||| | v 1 v 2 ··· v p b ||| | D

whose columns are the v i ’s and the b ’s.

Now we have (at least) two equivalent ways of thinking about systems of equations:

  1. Augmented matrices.
  2. Linear combinations of vectors (vector equations).

The second is more geometric in nature: it lends itself to drawing pictures.

Subsection3.2.2Spans

It will be important to know what are all linear combinations of a set of vectors v 1 , v 2 ,..., v p in R n . In other words, we would like to understand the set of all vectors b in R n such that the vector equation (in the unknowns x 1 , x 2 ,..., x p )

x 1 v 1 + x 2 v 2 + ··· + x p v p = b

has a solution (i.e. is consistent).

Definition

Let v 1 , v 2 ,..., v p be vectors in R n . The span of v 1 , v 2 ,..., v p is the collection of all linear combinations of v 1 , v 2 ,..., v p , and is denoted Span { v 1 , v 2 ,..., v p } . In symbols:

Span { v 1 , v 2 ,..., v p } = A x 1 v 1 + x 2 v 2 + ··· + x p v p | x 1 , x 2 ,..., x p in R B

We also say that Span { v 1 , v 2 ,..., v p } is the subset spanned by or generated by the vectors v 1 , v 2 ,..., v p .

The above definition is the first of several essential definitions that we will see in this textbook. They are essential in that they form the essence of the subject of linear algebra: learning linear algebra means (in part) learning these definitions. All of the definitions are important, but it is essential that you learn and understand the definitions marked as such.

Set Builder Notation

The notation

A x 1 v 1 + x 2 v 2 + ··· + x p v p | x 1 , x 2 ,..., x p in R B

reads as: “the set of all things of the form x 1 v 1 + x 2 v 2 + ··· + x p v p such that x 1 , x 2 ,..., x p are in R . The vertical line is “such that”; everything to the left of it is “the set of all things of this form”, and everything to the right is the condition that those things must satisfy to be in the set. Specifying a set in this way is called set builder notation.

All mathematical notation is only shorthand: any sequence of symbols must translate into a usual sentence.

Three characterizations of consistency

Now we have three equivalent ways of making the same statement:

  1. A vector b is in the span of v 1 , v 2 ,..., v p .
  2. The vector equation
    x 1 v 1 + x 2 v 2 + ··· + x p v p = b
    has a solution.
  3. The linear system with augmented matrix
    C ||| | v 1 v 2 ··· v p b ||| | D
    is consistent.

Equivalent means that, for any given list of vectors v 1 , v 2 ,..., v p , b , either all three statements are true, or all three statements are false.

Figure9This is a picture of an inconsistent linear system: the vector w on the right-hand side of the equation x 1 v 1 + x 2 v 2 = w is not in the span of v 1 , v 2 . Convince yourself of this by trying to solve the equation x 1 v 1 + x 2 v 2 = w by moving the sliders, and by row reduction. Compare this figure.
Pictures of Spans

Drawing a picture of Span { v 1 , v 2 ,..., v p } is the same as drawing a picture of all linear combinations of v 1 , v 2 ,..., v p .

Span { v } v Span { v , w } v w Span { v , w } v w
Figure10Pictures of spans in R 2 .
Span { v } v Span { v , w } v w v w u Span { u , v , w } Span { u , v , w } v w u
Figure13Pictures of spans in R 3 . The span of two noncollinear vectors is the plane containing the origin and the heads of the vectors. Note that three coplanar (but not collinear) vectors span a plane and not a 3-space, just as two collinear vectors span a line and not a plane.