Lecture 6: Corrections; Dimension; Linear maps

Transcription

1 Lecture 6: Corrections; Dimension; Linear maps Travis Schedler Tues, Sep 28, 2010 (version: Tues, Sep 28, 4:45 PM)

2 Goal To briefly correct the proof of the main Theorem from last time. (See website for details) To understand dimensions of subspaces and sums To introduce linear maps. Motivation: Understand relationships between vector spaces!

3 The fundamental inequality corrected Theorem (Theorem 2.6) The length of every linearly independent list is less than or equal to the length of every spanning list. I made a mistake in the proof last time: Given a lin. ind. list (u 1,..., u m ) and a spanning list (v 1,..., v n ). Idea: want {u 1,..., u m } {v 1,..., v n }. Last time I said: Make all the nonzero v j appear in (u 1,..., u m ). Problem: can t if the (v 1,..., v n ) are lin. dependent (and none are zero). Solution one: Can first replace (v 1,..., v n ) by a basis. Better: Just replace any u k / {v 1,..., v n } by any v j / Span(u 1,..., u k 1, u k+1,..., u m ). Eventually, {u 1,..., u m } {v 1,..., v n }. Since they are distinct, m n.

4 Corollary (Corollary more) Every finite-dimensional vector space V has a basis, and every basis has length dim V. Proof. Every spanning list of length dim V must be minimal, hence linearly independent, i.e., a basis. As explained last time, the theorem implies that all bases have the same length. Corollary (Theorem 2.12) Every linearly independent list in a finite-dimensional vector space can be extended to a basis. Proof. Linearly independent lists in V of length < dim V don t span, so can be extended. Linearly independent lists of length dim V must have length exactly dim V. Thus, they are maximal and span. If a linearly independent list already has length dim V, it is a basis.

5 Extending bases vs. direct sums Theorem Given any subspace U V, (i) Any basis (u 1,..., u m ) of U admits an extension (u 1,..., u m, w 1,..., w n ) to a basis of V. (ii) There exists a subspace W V such that V = U W. Proof: For (i), since (u 1,..., u m ) is linearly independent, it extends to a basis. For (ii), let W := Span(w 1,..., w n ).

6 Dimension and Subspaces Theorem (Propositions 2.7 and 2.15) If V is finite-dimensional and U V is a subspace, then U is finite-dimensional, and dim U dim V. Proof. Any linearly independent list in U is also linearly independent in V, so has length at most dim V. So, there is a maximal such list, which must be a basis. Since it has length at most dim V, we obtain that dim U dim V.

7 Infinite-dimensionality Conversely: If U V and U is infinite-dimensional, then so is V. Examples: P(R) R Cont. fns. R R All functions R R. So all infinite-dimensional! F n P(F), by polynomials of degree n 1. So dim(p(f)) n for all n, i.e., infinite-dimensional.

8 Dimension arithmetic Theorem (Theorem 2.18) Let U, W V. Then, dim(u + W ) = dim U + dim W dim(u W ). Proof. Take a basis of U W, (v 1,..., v k ), k = dim(u W ). Extend it to a basis of U: (v 1,..., v k, u 1,..., u m ), k + m = dim U. On the other hand, extend (v 1,..., v k ) to a basis of W : (v 1,..., v k, w 1,..., w n ), k + n = dim W. Thus, (u 1,..., u m, v 1,..., v k, w 1,..., w n ) spans U + W. Claim: it is lin. ind., hence a basis. This implies dim(u + W ) = m + k + n = (m + k) + (n + k) k = dim U + dim W dim(u W ).

9 More on dimensions Proposition (Proposition 2.19) Suppose V is finite-dimensional and U 1,..., U m V are subspaces such that V = U U m. Then, this sum is direct if and only if dim(v ) = dim(u 1 ) + + dim(u m ). (1.2) Proof. In the case m = 2, this was (almost) the warmup to last class! Generally, we show that the sum is direct if and only if, given bases of U 1,..., U m, put together they give a basis of V. Since, put together, they always span, they give a basis if and only if the size, dim U dim U m, equals dim V.

10 Linear maps Motivations: Explain the inclusions of polynomials, continuous functions, all functions, etc. Understand more general linear maps: Projections onto planes and lines Rotations, reflections Fourier transforms More general changes of bases Systems of equations!

11 Definition of linear maps Definition A linear map is a function T : V W between vector spaces V and W preserving addition and scalar multiplication. Explicitly, T is a function satisfying: T (u + v) = T (u) + T (v), for all u, v V, T (λv) = λt (v). Definition The set of linear maps V W is denoted L(V, W ). We will prove shortly that this is itself a vector space (and more).

12 Basic examples of linear maps The zero map 0 : V W : 0(v) = 0 for all v V. The identity map I : V V : I (v) = v for all v. Inclusions of subspaces: T : U V where U V. Here T (u) = u. Scalar multiplication: λi : V V : λi (v) = λv for all v. (Can combine with above!)

13 More sophisticated examples Differentiation of differentiable functions (R to R). E.g., x n nx n 1. Gives P(R) P(R), linear. Integration of continuous functions (R to R): b Definite integration: given a b, f f (y) dy R. Gives a C(R) R. x Indefinite integration: f f (y) dy, gives C(R) C(R). a Multiplication by a function: Fix a function f. This defines M f : functions functions, M f (g) := fg. For example, multiplication by 2x 3 : M 2x 3(1 + x) = 2x 3 + 2x 4. Backward shift: T : F F, T (x 1, x 2, x 3,...) = (x 2, x 3,...). Projections: Given V = U W, we obtain a map V U called projection: (u + w) u. Since all v can uniquely be written as v = u + w, the map v w is well-defined!