Advanced Linear Combinations & Systems of Linear Equations 1

1.4 Linear Combinations Math 4377/6308 Advanced Linear Algebra 1.4 Linear Combinations & Systems of Linear Equations Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 1 / 30 1.4 Linear Combinations Reduction Span Determining Subspaces 1.4 Linear Combinations & Systems of Linear Equations Linear Combinations: Definition Linear Combinations of Vectors in R2 Linear Combinations and Vector Equation Solving a System of Linear Equations by Row Eliminations Span of a Set of Vectors: Definition Span of a Set of Vectors in R2 and in R3 A Shortcut for Determining Subspaces Spanning Sets Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 2 / 30 1.4 Linear Combinations Reduction Span Determining Subspaces Linear Combinations Definition Let V be a vector space and S a nonempty subset of V . A vector v ∈ V is called a linear combination of vectors of S if there exist a finite number of vectors u1 , u2 , · · · , un in S and scalars a1 , a2 , · · · , an in F such that v = a1 u1 + a2 u2 + · · · + an un . In this case we also say that v is a linear combination of u1 , u2 , · · · , un and call a1 , a2 , · · · , an the coefficients of the linear combination Note that 0v = 0 for each v ∈ V , so the zero vector is a linear combination of any nonempty subset of V . Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 3 / 30 1.4 Linear Combinations Reduction Span Determining Subspaces Linear Combinations of Vectors in R2 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 vertices are 0, u and v. Geometric Description of R2   x1 Vector is the point x2 (x1 , x2 ) in the plane. R2 is the set of all points in the plane. Example     1 2 and v = . 3 1 Graphs of u, v and u + v are: Let u = Jiwen He, University of Houston (Parallelogram Rule) Math 4377/6308, Advanced Linear Algebra Spring, 2015 4 / 30 1.4 Linear Combinations Reduction Span Determining Subspaces Linear Combinations of Vectors in R2 (cont.) Example   1 Let u = . 2 Express u, 2u, and on a graph. Jiwen He, University of Houston −3 2 u Math 4377/6308, Advanced Linear Algebra Spring, 2015 5 / 30 1.4 Linear Combinations Reduction Span Determining Subspaces 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 Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 6 / 30 1.4 Linear Combinations Reduction Span Determining Subspaces 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 Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 7 / 30 1.4 Linear Combinations Reduction Span Determining Subspaces 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 Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra x1 = =⇒ x2 = x3 = Spring, 2015 8 / 30 1.4 Linear Combinations Reduction Span Determining Subspaces 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  Jiwen He, University of Houston  a1 a2 a3 b . Math 4377/6308, Advanced Linear Algebra Spring, 2015 9 / 30 1.4 Linear Combinations Reduction Span Determining Subspaces 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. Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 10 / 30