Rank and linear transformations

Transcription

1 Rank and linear transformations Math 4, Introduction to Linear Algebra Friday, February, Recall from Wednesday... Important characteristic of a basis Theorem. Given a basis B {v,...,v k } of subspace S, there is a unique way to epress any v S as a linear combination of basis vectors v,...,v k. Proof sketch. Suppose span(b) S v c v + c v + + c k v k for scalars c i,d i v d v + d v + + d k v k Then (c d )v +(c d )v + +(c k d k )v k c d c d.. c k d k c i d i i linearly independent

2 Recall from Wednesday... Eample of matri subspaces bases A A EROs R 3 4 basis for row(a) 3, 3 4 basis for col(a) 6, 7 Eample of matri subspaces bases A , 4, 5 free 3 A EROs R 3 4 basis for null(a) 3, 3,

3 Dimensions of fundamental subspaces For the 3 5 matri from the last eample, we have basis for row(a) 3, basis for 6, 7 basis for col(a) null(a), 3, 4 Note that dim(row(a)) dim(col(a)) dim(null(a)) 3 row rank of A column rank of A nullity of A Question: Is it a coincidence that row rank of A column rank of A? Question: Is it a coincidence that nullity of A + row/col rank of A # of cols of A? Rank of a matri Theorem. For any matri A, # of nonzero rows of RREF of A row rank of A column rank of A. rank of A # of leading s in RREF of A Consequence: rank(a) rank(a T ) Theorem (Rank Theorem). For any m n matri A, nullity(a) + rank(a) n. # of columns without leading s in RREF of A # of columns with leading s in RREF of A

4 Fundamental Theorem of Invertible Matrices (etended) Theorem. Let A be an n n matri. The following statements are equivalent: A is invertible. A b has a unique solution for all b. A has only the trivial solution. The RREF of A is I. A is the product of elementary matrices. rank(a) n. nullity(a). Columns of A are linearly independent. Columns of A span. Columns of A form a basis for. Rows of A are linearly independent. Rows of A span. Rows of A form a basis for. Eamples of vector spaces and subspaces R 3, under vector addition and scalar multiplication (scalars: R) S a b : a, b R vector spaces, under vector addition and scalar multiplication (scalars: R) span (v,v,...,v k ) subspaces

5 Thinking beyond Euclidean vectors: more eamples of vector spaces and subspaces set of m n matrices with entries from C, under matri addition and scalar multiplication (scalars: C) set of n n diagonal matrices set of all functions from R to R, under function addition and scalar multiplication (scalars: R) vector spaces set of all differentiable functions set of all continuous functions set of all polynomials of degree at most 3 with real coefficients, with addition and scalar mult. defined in usual way (scalars: R) set of all polynomials of degree at most w/real coefficients subspaces A central idea of linear algebra: linear transformations Definition A function T : R m is a linear transformation if u,v and scalars c, T (u + v) T (u)+t (v), and T (cv) ct(v). { streamlined as T (c u + c v) c T (u)+c T (v) Eample T 6 T T : R 3 R and T y z domain T 3 codomain R 3 3 y T R

6 A central idea of linear algebra: linear transformations Definition A function T : R m is a linear transformation if u,v and scalars c, T (u + v) T (u)+t (v), and T (cv) ct(v). { streamlined as T (c u + c v) c T (u)+c T (v) Non-eample T : R R and T y + y y Consider T T 4 4 T Kernel and range of linear transformation Let T : R m be a linear transformation. Definition The kernel or null space of T is ker(t ){v : T (v) }. set of all images under T The image or range of T is range(t ){ w R m : w T (v) for some v }. T R m ker(t) range(t)

7 Looking closer at an eample Eample T : R 3 R and T y z y ker(t ) v R 3 : T (v) : z R span z For any,y R, T y y so range(t )R subspaces! Remarks on linear transformations T R m ker(t) range(t) For any linear transformation T, T(), ker(t) is a subspace of, R m range(t) is a subspace of. We define nullity(t) dim(ker(t)) rank(t) dim(range(t)) Theorem (Rank Thm for linear transformations). For a linear transformation T : R m, rank(t ) + nullity(t ) dim( )n.

8 Matri multiplication as a linear transformation Primary eample of a linear transformation matri multiplication Given an m n matri A, define T () A for. Then T is a linear transformation. Astounding! Matri multiplication defines a linear transformation. This new perspective gives a dynamic view of a matri (it transforms vectors into other vectors) and is a key to building math models to physical systems that evolve over time (so-called dynamical systems). Matri multiplication as a mapping (or function) Given an m n matri A, define T () A for. Verify that T is a linear transformation. We have T (u + v) A(u + v) Au + Av T (u)+t(v) and T (cv) A(cv) c(av) ct(v) What is range of T? range(t ) set of all vectors b R m such that T () b for some set of all vectors b R m such that A b for some col(a) Similarly, ker(t ) null(a)

9 A linear transformation as matri multiplication Theorem. Every linear transformation T : R m can be represented by an m n matri A so that, T () A. More astounding! Question Given T, how do we find A? Transformation T is completely determined by its action on basis vectors. Consider standard basis vectors for : e.,...,e n. Compute T (e ),T(e ),...,T(e n ). Standard matri of a linear transformation Question Given T, how do we find A? Transformation T is completely determined by its action on basis vectors. Consider standard basis vectors for : e.,...,e n. Compute T (e ),T(e ),...,T(e n ). Then A T (e ) T (e ) T (e n ) is called the standard matri for T.

10 Standard matri for an eample Eample T : R 3 R and T y y z T T T A A 5 What is T 5? 5 5