Vector Equation

1.3 Vector Equations Math 2331 – Linear Algebra 1.3 Vector Equations Math 2331, Linear Algebra 1 / 18 1.3 Vector Equations Definition Combinations Span 1.3 Vector Equations Vector in R2 Geometric Description of R2 Parallelogram Rule Vectors in Rn Linear Combinations Example: Linear Combinations of Vectors in R2 Vector Equation Span of a Set of Vectors: Definition Spanning Sets in R3 Geometric Description of Span{v } Geometric Description of Span{u, v } Math 2331, Linear Algebra 2 / 18 1.3 Vector Equations Definition Combinations Span Vector Key Concepts to Master linear combinations of vectors and a spanning set. Vectors in Rn    vectors with n entries: u =   u1 u2 .. .     , a matrix with one column.  un Geometric Description of R2   x1 Vector is the point (x1 , x2 ) in the plane. x2 R2 is the set of all points in the plane. Math 2331, Linear Algebra 3 / 18 1.3 Vector Equations Definition Combinations Span Parallelogram Rule Parallelogram Rule for Addition of Two Vectors If u and v in R2 are represented as points in the plane, then u + v corresponds to the fourth vertex of the parallelogram whose other   0 vertices are 0, u and v. (Note that 0 = .) 0 Example     1 2 Let u = and v = . 3 1 Graphs of u, v and u + v are: (Parallelogram Rule) Math 2331, Linear Algebra 4 / 18 1.3 Vector Equations Definition Combinations Span 2 Vectors in R : Example Example   1 Let u = . 2 Express u, 2u, and on a graph. −3 2 u Math 2331, Linear Algebra 5 / 18 1.3 Vector Equations Definition Combinations Span Linear Combinations of Vectors Linear Combinations of Vectors Given vectors v1 , v2 , . . . , vp in Rn and given scalars c1 , c2 , . . . , cp , the vector y defined by y = c1 v1 + c2 v2 + · · · + cp vp is called a linear combination of v1 , v2 , . . . , vp using weights c 1 , c2 , . . . , cp . Examples (Linear Combinations of v1 and v2 ) 3v1 + 2v2 , 1 3 v1 , v1 − 2v2 , Math 2331, Linear Algebra 0 6 / 18 1.3 Vector Equations Definition Combinations Span Linear Combinations of Vectors in R2 : Example Example     2 −2 Let v1 = and v2 = . Express each of the 1 2 following as a linear combination of v1 and v2 :         0 −4 6 7 a= , b= , c= , d= 3 1 6 −4 Math 2331, Linear Algebra 7 / 18 1.3 Vector Equations Definition Combinations Span Linear Combinations: Example Example    1 Let a1 =  0 , a2 =  3 Determine if b is a linear      4 3 −1 2 , a3 =  6 , and b =  8 . 14 10 −5 combination of a1 , a2 , and a3 . Solution: Vector b is a linear combination of a1 , a2 , and a3 if can we find weights x1 , x2 , x3 such that x1 a1 + x2 a2 + x3 a3 = b. Vector Equation (fill-in):         1 4 3 −1 x1  0  + x2  2  + x3  6  =  8  3 14 10 −5 Math 2331, Linear Algebra 8 / 18 1.3 Vector Equations Definition Combinations Span Linear Combinations: Example (cont.) Corresponding System: x1 3x1 4x2 + 3x3 = −1 2x2 + 6x3 = 8 + 14x2 + 10x3 = −5 + Corresponding Augmented Matrix:     1 4 3 −1 1 0 0 1  0 2 6 8  ∼  0 1 0 −2  3 14 10 −5 0 0 1 2 Math 2331, Linear Algebra x1 = =⇒ x2 = x3 = 9 / 18 1.3 Vector Equations Definition Combinations Span Linear Combinations: Review Review of the last example: a1 , a2 , a3 and b are columns of the augmented matrix  1  0 3 ↑ a1 4 3 2 6 14 10 ↑ ↑ a2 a3  −1 8  −5 ↑ b Solution to x1 a1 + x2 a2 + x3 a3 = b is found by solving the linear system whose augmented matrix is   a1 a2 a3 b . Math 2331, Linear Algebra 10 / 18 1.3 Vector Equations Definition Combinations Span Linear Combinations and Vector Equation Vector Equation A vector equation x1 a1 + x2 a2 + · · · + xn an = b has the same solution set as the linear system whose augmented matrix is   a1 a2 · · · an b . In particular, b can be generated by a linear combination of a1 , a2 , . . . , an if and only if there is a solution to the linear system corresponding to the augmented matrix. Math 2331, Linear Algebra 11 / 18 1.3 Vector Equations Definition Combinations Span Span of a Set of Vectors: Examples Example   3 Let v =  4  . 5   0 Label the origin  0  0 together with v, 2v and 1.5v on the graph. v, 2v and 1.5v all lie on the same line. Span{v} is the set of all vectors of the form cv. Here, Span{v} = a line through the origin. Math 2331, Linear Algebra 12 / 18 1.3 Vector Equations Definition Combinations Span Span of a Set of Vectors: Examples (cont.) Example Label u, v, u + v and 3u+4v on the graph. u, v, u + v and 3u+4v all lie in the same plane. Span{u, v} is the set of all vectors of the form x1 u + x2 v. Here, Span{u, v} = a plane through the origin. Math 2331, Linear Algebra 13 / 18 1.3 Vector Equations Definition Combinations Span Span of a Set of Vectors: Definition Span of a Set of Vectors Suppose v1 , v2 , . . . , vp are in Rn ; then Span{v1 , v2 , . . . , vp } = set of all linear combinations of v1 , v2 , . . . , vp . Span of a Set of Vectors (Stated another way) Span{v1 , v2 , . . . , vp } is the collection of all vectors that can be written as x1 v1 + x2 v2 + · · · + xp vp where x1 , x2 , . . . , xp are scalars. Math 2331, Linear Algebra 14 / 18 1.3 Vector Equations Definition Combinations Span Span of a Set of Vectors: Example Example  Let v1 = 2 1   and v2 = 4 2  . (a) Find a vector in Span{v1 , v2 }. (b) Describe Span{v1 , v2 } geometrically. Math 2331, Linear Algebra 15 / 18 1.3 Vector Equations Definition Combinations Span 3 Spanning Sets in R Example v2 is a multiple of v1 Span{v1 , v2 } =Span{v1 } =Span{v2 } (line through the origin) Math 2331, Linear Algebra 16 / 18 1.3 Vector Equations Definition Combinations Span 3 Spanning Sets in R (cont.) Example   4 Let v1 =  2  2   6 and v2 =  3 . 3 Is Span{v1 , v2 } a line or a plane? v2 is not a multiple of v1 Span{v1 , v2 } =plane through the origin Math 2331, Linear Algebra 17 / 18 1.3 Vector Equations Definition Combinations Span Spanning Sets Example  1 Let A =  3 0 spanned by the Solution:    2 8 1  and b =  3 . Is b in the plane 5 17 columns of A? ? Do x1 and x2 exist so that       1 2 8 x1  3  + x2  1  =  3  0 5 17 Corresponding augmented matrix:       1 2 8 1 2 8 1 2 8  3 1 3  ∼  0 −5 −21  ∼  0 −5 −21  0 5 17 0 5 17 0 0 −4 So b is not in the plane spanned by the columns of A Math 2331, Linear Algebra 18 / 18