Linear Independence Linear Algebra MATH 76 Linear Algebra LI or LD Chapter, Section 7 /
Linear Combinations and Span Suppose s, s,..., s p are scalars and v, v,..., v p are vectors (all in the same space R n ). We call s v + s v + + s p v p a linear combination of the vectors v, v,..., v p. We always have the trivial linear combination v + v + + v p =. Here we want to know when there is a non-trivial LC of v, v,..., v p that equals. This means that s v + s v + + s p v p = and some scalar s j. Linear Algebra LI or LD Chapter, Section 7 /
The span of v, v,..., v p Span{ v, v,..., v p } = { w w = s v + s v + + s p v p, s j scalars } The span of a single vector is a line. Except when this is not true. The span of a two vectors is: a line if the two vectors are parallel a plane if the two vectors are not parallel except when this is not true What is the span of 7 vectors? It depends.... Linear Algebra LI or LD Chapter, Section 7 /
Linear Dependence The vectors v, v,..., v p are linearly dependent if there is a non-trivial LC of them that equals : that is, if there are scalars s, s,..., s p so that s v + s v + + s p v p =, and (at least) one of the scalars is non-zero. Linearly dependent vectors carry redundant information. Vectors that are not LD are said to be linearly independent. Linearly independent vectors carry NO redundant information. Linear Algebra LI or LD Chapter, Section 7 4 /
Example three vectors in R Are the vectors 4 LD or LI? Notice that + = 4. Thus + 4 =. So the three vectors are LD. This means that Span{ 4 } = Span{ } which is a plane in R. Linear Algebra LI or LD Chapter, Section 7 5 /
Example three vectors in R Are the vectors 7 LD or LI? Well, + 5 = 7, so + 5 7 =. Therefore, the three vectors are LD. This means that Span{ 7 } = Span{ } which is a plane in R. Linear Algebra LI or LD Chapter, Section 7 6 /
Example three vectors in R Are the vectors LD or LI? Since + =, we get + =. So the three vectors are LD. This means that Span{ } = Span{ } which is a plane in R. Linear Algebra LI or LD Chapter, Section 7 7 /
Example three vectors in R 4 Are the vectors v = v = v = 6 LD or LI? Easy to see that v = v v, so v v v = and therefore v, v, v are LD. Also, Span { } { } v, v, v = Span v, v which is a -plane in R 4. But what if we had five (or fifteen) vectors? Let A = [ ] v v v be the matrix with columns given by v, v, v. Recall that A x is the LC of the columns of A given by x A x = A x = x v + x v + x v. x See that there is a non-trivial LC x v + x v + x v = if and only if there is a non-trivial (i.e., non-zero) solution to A x =. Linear Algebra LI or LD Chapter, Section 7 8 /
Example three vectors in R 4 (continued) We row reduce A to get 6 R R R 4 R R + R R 4 R Since the last column has no row leader, we know that x is a free variable. Thus x = t (with t any scalar), x = t and x = t. There are infinitely many solutions to A x =, and all but one of these is non-trivial. Thus, there are infinitely many non-trival LCs s v + s v + s v =. Linear Algebra LI or LD Chapter, Section 7 9 /
Linearly Dependent Vectors Suppose the vectors v, w are LD. This means there are scalars s, t such that s v + t w = and s, t are NOT both zero. Assume s. Then we can write v = (t/s) w. Thus v is a scalar multiple of w; i.e., v is a LC of w. Suppose v is a LC of v, v,..., v p. This means there are scalars s, s,..., s p so that v = s v + s v + + s p v p. Then v ( s v + s v + + s p v p ) =. so we see that v, v, v,..., v p are LD. If some v j is an LC of the other vectors, then v, v, v,..., v p are LD. Converse true too! Linear Algebra LI or LD Chapter, Section 7 /
A useful Fact too many vectors are always LD If p > n, then any set of p vectors in R n is LD. Any 5 vectors in R 4 are LD. Any vectors in R 7 are LD. Why? Just look at REF of appropriate coefficient matrix! Linear Algebra LI or LD Chapter, Section 7 /
Parallel Vectors Two vectors are parallel if and only if one is a scalar multiple of the other. Any two non-zero vectors are LI if and only if they are not parallel. Why? (a good quiz question!) Suppose we have three vectors v, v, v (say, in R 5 ) and v v v v. Are v, v, v LD or LI? Hint: it is not difficult to produce a simple example! (a good exam question ) Linear Algebra LI or LD Chapter, Section 7 /