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How to Draw Plane Given the Span

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Objectives
  1. Sympathise the equivalence betwixt a organization of linear equations and a vector equation.
  2. Learn the definition of Span { x 1 , x two ,..., ten k } , and how to depict 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 consequent system of equations, spans in R ii and R 3 .
  5. Vocabulary word: vector equation.
  6. Essential vocabulary word: span.

An equation involving vectors with north coordinates is the same as due north equations involving only numbers. For instance, the equation

x C ane 2 6 D + y C − 1 − 2 − 1 D = C 8 16 3 D

simplifies to

C x 2 x six ten D + C − y − 2 y − y D = C eight 16 3 D or C ten − y ii 10 − 2 y 6 ten − y D = C viii xvi three D .

For ii vectors to exist equal, all of their coordinates must be equal, so this is just the organisation of linear equations

E ten − y = viii 2 x − two y = xvi half-dozen 10 − y = three.

Definition

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

Asking whether or non a vector equation has a solution is the same every bit request if a given vector is a linear combination of some other given vectors.

For instance the vector equation above is request if the vector ( 8,16,three ) is a linear combination of the vectors ( 1,two,6 ) and ( − 1,ii, − 1 ) .

The thing nosotros really care about is solving systems of linear equations, not solving vector equations. The whole bespeak of vector equations is that they give usa a different, and more geometric, style of viewing systems of linear equations.

Figure ivA picture of the to a higher place vector equation. Try to solve the equation geometrically by moving the sliders.

In order to actually solve the vector equation

x C i 2 6 D + y C − 1 − 2 − 1 D = C viii sixteen 3 D ,

one has to solve the system of linear equations

E x − y = viii 2 x − 2 y = 16 6 10 − y = 3.

This means forming the augmented matrix

C 1 − 1 8 2 − ii 16 6 − 1 3 D

and row reducing. Annotation that the columns of the augmented matrix are the vectors from the original vector equation, so it is not really necessary to write the system of equations: one tin go directly from the vector equation to the augmented matrix by "smooshing the vectors together". In the following instance we carry out the row reduction and find the solution.

Example

Recipe: Solving a vector equation

In full general, the vector equation

x ane v i + x 2 v 2 + ··· + x grand v k = b

where 5 1 , v 2 ,..., 5 k , b are vectors in R n and x 1 , ten 2 ,..., x k are unknown scalars, has the same solution set as the linear organization with augmented matrix

C ||| | 5 i 5 2 ··· v g b ||| | D

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

Now we have three equivalent means of thinking near a linear system:

  1. Every bit a arrangement of equations:

    H 2 10 1 + three 10 2 − two 10 three = vii x i − x 2 − iii ten 3 = 5

  2. As an augmented matrix:

    F 23 − two 7 1 − 1 − 3 5 G

  3. Equally a vector equation ( 10 1 5 1 + 10 2 v two + ··· + ten n 5 due north = b ):

    x 1 F 2 1 G + x 2 F 3 − 1 G + x 3 F − 2 − 3 Grand = F 7 5 G

The third is geometric in nature: information technology lends itself to drawing pictures.

It will be important to know what are all linear combinations of a gear up of vectors v 1 , 5 2 ,..., five k in R n . In other words, we would like to understand the prepare of all vectors b in R northward such that the vector equation (in the unknowns 10 1 , ten two ,..., x chiliad )

x ane v one + x 2 v two + ··· + x k v thousand = b

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

Definition

Let 5 ane , v 2 ,..., v k exist vectors in R northward . The span of v ane , five 2 ,..., v k is the drove of all linear combinations of v 1 , v ii ,..., 5 grand , and is denoted Span { v 1 , 5 2 ,..., v k } . In symbols:

Bridge { v 1 , v 2 ,..., 5 k } = A x 1 5 ane + ten 2 v 2 + ··· + x 1000 five k | x 1 , x two ,..., ten k in R B

We also say that Span { 5 1 , five 2 ,..., 5 k } is the subset spanned by or generated by the vectors v i , v 2 ,..., 5 yard .

The above definition is the commencement of several essential definitions that we volition encounter in this textbook. They are essential in that they form the essence of the subject area of linear algebra: learning linear algebra means (in part) learning these definitions. All of the definitions are important, but it is essential that y'all larn and sympathise the definitions marked equally such.

Equivalent means that, for whatsoever given list of vectors v 1 , v 2 ,..., v g , b , either all three statements are true, or all three statements are simulated.

Figure 10This is a motion picture of an inconsistent linear system: the vector due west on the right-paw side of the equation x one 5 1 + x 2 5 2 = w is non in the bridge of v 1 , v ii . Convince yourself of this by trying to solve the equation 10 i v 1 + ten 2 five two = westward by moving the sliders, and past row reduction. Compare this figure.
Pictures of Spans

Drawing a picture of Bridge { v ane , v 2 ,..., v k } is the same as drawing a picture of all linear combinations of v i , v 2 ,..., v one thousand .

Span { five } 5 Span { 5 , w } v w Span { five , w } v w

Effigy xiPictures of spans in R 2 .

Span { 5 } v Bridge { v , w } v w 5 w u Span { u , v , w } Bridge { u , v , w } v w u

Figure 12Pictures of spans in R 3 . The span of ii noncollinear vectors is the airplane containing the origin and the heads of the vectors. Notation that iii coplanar (simply not collinear) vectors span a plane and not a three-space, just as two collinear vectors span a line and not a plane.

Interactive: Span of two vectors in R ii

Interactive: Span of two vectors in R iii

Interactive: Span of three vectors in R iii

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Source: https://textbooks.math.gatech.edu/ila/spans.html

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