Linear Independence MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015
Introduction Given a set of vectors {v 1, v 2,..., v r } and another vector v span{v 1, v 2,..., v r } we have seen that we can write v as a linear combination of {v 1, v 2,..., v r }, i.e., v = c 1 v 1 + c 2 v 2 + + c r v r. Remark: the choice of the coefficients c 1, c 2,..., c r may not be unique. Today we will explore conditions under which the coefficients are unique.
Linear Independence (1 of 2) Definition If S = {v 1, v 2,..., v r } is a set of two or more vectors in a vector space V, then S is said to be linearly independent provided no vector in S can be expressed as a linear combination of the others. Otherwise S is a linearly dependent set.
Linear Independence (1 of 2) Definition If S = {v 1, v 2,..., v r } is a set of two or more vectors in a vector space V, then S is said to be linearly independent provided no vector in S can be expressed as a linear combination of the others. Otherwise S is a linearly dependent set. Remark: if set S contains only one vector, then S is linearly independent if and only if that vector is nonzero.
Linear Independence (2 of 2) Theorem If S = {v 1, v 2,..., v r } is a nonempty set of vectors in vector space V, then the set S is linearly independent if and only if implies k 1 = k 2 = = k r = 0. k 1 v 1 + k 2 v 2 + + k r v r = 0
Examples Determine if the following sets are linearly independent in their respective vector spaces. Determine if {(1, 1), (1, 2)} R 2 is a linearly independent set. Determine if {(1, 1, 1), (1, 1, 0), (1, 0, 0)} R 3 is a linearly independent set. Determine if {(1, 2, 4), (2, 1, 3), (3, 3, 7)} R 3 is a linearly independent set. Determine if {1, x, x 2,..., x n } P n form a linearly independent set.
Linear Independence (continued) Theorem A set S with two or more vectors is: 1. linearly dependent if and only if at least one of the vectors of S is expressible as a linear combination of the other vectors in S. 2. linearly independent if and only if no vector of S is expressible as a linear combination of the other vectors in S.
Linear Independence (continued) Theorem A set S with two or more vectors is: 1. linearly dependent if and only if at least one of the vectors of S is expressible as a linear combination of the other vectors in S. 2. linearly independent if and only if no vector of S is expressible as a linear combination of the other vectors in S. Example Express one of the vectors in {(1, 2, 4), (2, 1, 3), (3, 3, 7)} as a linear combination of the other two.
Linear Independence (continued) Theorem 1. A finite set of vectors containing the zero vector is linearly dependent. 2. A set with exactly one vector is linearly independent if and only if that vector is not 0. 3. A set with exactly two vectors is linearly independent if and only if neither vector is a scalar multiple of the other.
Proof For any vectors v 1, v 2,..., v r let S = {0, v 1, v 2,..., v r }. Note that 0 = (1)0 + (0)v 1 + (0)v 2 + + +(0)v r and not all of the vector coefficients are zero.
Examples Are x and x 2 linearly independent as functions in F(, )?
Examples Are x and x 2 linearly independent as functions in F(, )? Are sin 2x and sin x cos x linearly independent as functions in F(, )?
Geometric Interpretation in R 2 Question: which of the following illustrations depicts linearly independent vectors? 1 y y 1 0.8 0.6 0.5 0.4 0.2-0.4-0.2 0.2 0.4 0.6 0.8 1 x -0.4-0.2 0.2 0.4 0.6 0.8 1-0.2 x -0.5-0.4-1
Geometric Interpretation in R 3 Question: which of the following illustrations depict(s) linearly independent vectors? 0.5 y 0.25 0-0.25 y 0 0.5-0.5-0.5 1 1 0.4 z 0.2 0-0.5-0.25 0 x 0.25 0.5 0.5 0.25 0 y -0.25-0.5 0.75 z 0.5 0.25 0-0.5-0.25 0 x 0.25 0.5 0.75 z0.5 0.25 0-0.5 0 x 0.5
Linear Independence (continued) Theorem Let S = {v 1, v 2,..., v r } be vectors from R n. If r > n, then S is linearly dependent.
Proof Suppose that v 1 = (v 11, v 12,..., v 1n ) v 2 = (v 21, v 22,..., v 2n ). v r = (v r1, v r2,..., v rn ) and consider the equation k 1 v 1 + k 2 v 2 + + k r v r = 0.
Proof Suppose that and consider the equation v 1 = (v 11, v 12,..., v 1n ) v 2 = (v 21, v 22,..., v 2n ). v r = (v r1, v r2,..., v rn ) k 1 v 1 + k 2 v 2 + + k r v r = 0. Writing the equation component-wise yields the system of equations: v 11 k 1 + v 21 k 2 + + v r1 k r = 0 v 12 k 1 + v 22 k 2 + + v r2 k r = 0 v 1n k 1 + v 2n k 2 + + v rn k r = 0.
Proof Suppose that and consider the equation v 1 = (v 11, v 12,..., v 1n ) v 2 = (v 21, v 22,..., v 2n ). v r = (v r1, v r2,..., v rn ) k 1 v 1 + k 2 v 2 + + k r v r = 0. Writing the equation component-wise yields the system of equations: v 11 k 1 + v 21 k 2 + + v r1 k r = 0 v 12 k 1 + v 22 k 2 + + v r2 k r = 0 v 1n k 1 + v 2n k 2 + + v rn k r = 0 This homogeneous system has r > n unknowns and thus.
Linear Independence of Functions Question: how do we determine if a (possibly large) set of functions is linearly dependent or independent?
Linear Independence of Functions Question: how do we determine if a (possibly large) set of functions is linearly dependent or independent? Answer: if the functions are S = {f 1, f 2,..., f n } where f i : R R is (n 1)-times differentiable, then we can use the Wronskian: W (x) = f 1 (x) f 2 (x) f n (x) f 1 (x) f 2 (x) f n(x)... f (n 1) 1 (x) f (n 1) 2 (x) f (n 1) n (x)
Wronskian Theorem If functions f 1, f 2,..., f n have n 1 continuous derivatives for x R then the functions are linearly independent in C n 1 (R) if there exists x R such that det(w (x)) 0.
Wronskian Theorem If functions f 1, f 2,..., f n have n 1 continuous derivatives for x R then the functions are linearly independent in C n 1 (R) if there exists x R such that det(w (x)) 0. Remark: the converse of the theorem is false. If det(w (x)) = 0 for all x R, no conclusion can be reached about the linear independence of f 1, f 2,..., f n.
Examples Determine if the following sets of functions are linearly dependent or independent on R. {e x, e x } {1, x, x 2, x 3 } {1, sin 2 x, cos 2 x}
Homework Read Section 4.3 Exercises: 1, 2, 7, 8, 10, 11, 13, 17, 20.