New concepts: Span of a vector set, matrix column space (range) Linearly dependent set of vectors Matrix null space

Transcription

1 Lesson 6: Linear independence, matrix column space and null space New concepts: Span of a vector set, matrix column space (range) Linearly dependent set of vectors Matrix null space

2 Two linear systems: same system matrix, dierent right hand side 2 2 R ; b 2 R ; c 2 R () x + 2x 2 + x = x 2 + x = x + 2x 2 + x = 8, x = b; y + 2y 2 + y = y 2 + y = y + 2y 2 + y =4, y = c Form the bordered matrix in both cases, and reduce to triangular form (Gaussian 2 2 8, x + 2x 2 + x = x 2 + x = = Innite number of , y + 2y 2 + 2y = y 2 = =/ No solutions

3 Interpret previous examples Recall: Solving x = b means nding the linear combination of columns of such that x a + x 2 a 2 + x a = b; x = ( x x 2 x ) T (2) For previous examples 2 2 ) a ; a2 2 2 ; a Note that a = a + a 2, so the linear combination (2) can be rewritten as (x + x )a + (x 2 + x )a 2 = b If b is in the plane dened by the two directions a ; a 2 then there are an innity of choices of x = ( x x 2 x ) T to obtain b by the linear combination of a ; a 2. This is the rst example. Since c = ( 4 ) T is not in the planed dened by a ; a 2, there is no linear combination of a ; a 2 to obtain c, hence there is no solution to the system y = c.

4 Span of a set of vectors, range of a matrix Generalize this idea of vectors reachable by linear combination of other vectors Denition. The span of vectors a ; a 2 ; :::; a n 2 V; is the set of vectors reachable by linear combination spanfa ; a 2 ; :::; a n g = fb 2 V j 9x ; :::; x n 2 S such that b = x a + :::x n a n g: The notation used for set on the right hand side is read: those vectors b in V with the property that there exist n scalars x ; :::; x n to obtain b by linear combination of a ; a 2 ; :::; a n. linear combination is conveniently expressed as a matrix-vector product leading to a dierent formulation of the same concept Denition. The column space (or range) of matrix 2 R mn is the set of vectors reachable by linear combination of the matrix column vectors C() = range() = fb 2 R m j 9x 2 R n such that b = xg R m

5 Linearly dependent vectors In the example () = ( a a 2 a ) 2 2 spanfa ; a 2 ; a g = spanfa ; a 2 g since a = a + a 2, a + a 2 a =. Introduce a concept to capture the idea that a vector can be expressed in terms of other vectors. Denition. The vectors a ; a 2 ; :::; a n 2 V ; are linearly dependent if there exist n scalars, x ; :::; x n 2 S, at least one of which is dierent from zero such that x a + :::x n a n = Note that fg, with 2 V is a linearly dependent set of vectors since =.

6 Linearly independent vectors The converse of linear dependence is linear independence, a member of the set cannot be expressed as a non-trivial linear combination of the other vectors Denition. The vectors a ;a 2 ;:::;a n 2V;are linearly independent if the only n scalars, x ;:::; x n 2 S, that satisfy are x =, x 2 =,...,x n =. x a + :::x n a n = ; () The choice x = ( x ::: x n ) T = that always satises () is called a trivial solution. We can restate linear independence as () being satised only by the trivial solution.

7 Null space Introduce a characterization of the column vectors of a matrix related to linear dependence Denition. The null space of a matrix 2 R mn is the set N() = null() = fx 2 R n jx = g R n If null() = fg then the column vectors of are linearly independent, since the only way to satisfy () is by the trivial solution x = For example () we have c (a + a 2 a ) = for any scalar c, hence 2 2 ) a ; a2 2 2 ; a 8 < C() = spanfa ; a 2 g; N() = : 9 = ;