Given a set of vectors, how to determine if there are any vectors that are linear combinations of other vectors? Idea: in cfw_u1, , ui, , uk, if is ui a linear combination of other vectors, then there exists scalars c1, , ci, , ck, not all zero, + ckuk =
Span of a set of vectors
Properties: Spancfw_u = the set of all multiple of u, and Spancfw_0 = cfw_0. S contains a nonzero vector. SpanS has infinitely many vectors. Example: :
SpanS3 = SpanS4 = R2
nonparallel vectors
Example: Spancfw_e1,e2 = xy-plane in
Gaussian elimination: an algorithm for finding a (actually the) reduced row echelon form of a matrix.
previous [A b]
pivot position
pivot column
interchange rows 1 and 2
pivot position
pivot column
1
pivot position : a row echelon form pivot column
multip
Linear equation
variables constant term coefficients
System of linear equations (m equations, n variables)
two linear equations in two variables (two lines in R2):
no solution
unique solution
infinitely many solution
(Will be proved subsequently.)
Solutio
Example: Given the coefficients (cfw_3,4,1), it is easy to compute the combination ([2 8]T), but the inverse problem is harder. Example: To determine x1 and x2, must solve a system of linear equations, which has a unique solution [x1 x2]T = [1 2]T in this
LINEAR ALGEBRA
This course will cover: 1. Matrices, Vectors, and Systems of Linear Equations
1.1 Matrices and vectors 1.2 Linear combination, matrix-vector products, and special matrices 1.3 Systems of linear equations 1.4 Gaussain elimination 1.6 The spa