Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.; More geerally, what does it mea that F ad F{0, 1,..., 1 } have dimesio? What does it mea to be ifiite-dimesioal? To uderstad the relatioship betwee vector spaces ad special spaig lists. Mai goals: dimesio. To uderstad spa, liear idepedece, bases, ad Spa Fix a field F throughout. Recall from last time ad the homework: Defiitio 1. Let v 1,..., v V. Spa(v 1,..., v ) = {λ 1 v 1 + + λ v λ 1,..., λ F}. Defiitio 2. A vector space V is fiite-dimesioal if V = Spa(v 1,..., v ) for some v 1,..., v V ad some 1. Defiitio 3. The dimesio dim(v ) of a fiite-dimesioal vector space is the miimum such that V = Spa(v 1,..., v ). Fiite-dimesioal examples Let V = R 3. The V = Spa((1, 0, 0), (0, 1, 0), (0, 0, 1)). So dim(v ) 3. If V = Spa(v 1,..., v ), the also V = Spa(v 1,..., v, 0). Also, for all v V, V = Spa(v 1,..., v, v). 1
We have F = Spa(δ 1,..., δ ) where δ i = (0, 0,..., 0, 1, 0, 0,..., 0), with 1 i the i-th coordiate ad 0 elsewhere. So, dim(f ). Similarly, F{0, 1,..., 1 } = Spa(δ 0, δ 1,..., δ 1 ). So, dim(f{0, 1,..., 1 }). Ifiite-dimesioal examples The vector space of polyomials P(F). Cotiuous fuctios [0, 1] R. Ifiite lists F. F(X) where X is ifiite. We ca prove the first directly (x d+1 does ot appear i ay fiite list of polyomials of maximum degree d). Secod oe: homework problem (!) Last two: There are liearly idepedet lists of arbitrary legth. We will show that, if V is fiite-dimesioal, the liearly idepedet lists are at most dim V i size. Liear (i)depedece Recall from last time: Defiitio 4. A liearly idepedet list of vectors (v 1,..., v ) is oe such that, if λ 1 v 1 + + λ v = 0, the λ 1 = = λ = 0. A list which is ot liearly idepedet is liearly depedet. The, λ 1 v 1 + + λ v = 0 for some λ i which are ot all zero. Examples: ((v 1,..., v, 0)) is always liearly depedet. For every vector space V ad ozero vector v V, (v) is liearly idepedet. I R 2, ((1, 0), (1, 1)) is liearly idepedet. I F, the list ((1, 0, 0,..., 0), (0, 1, 0,..., 0),..., (0, 0,..., 0, 1)) is liearly idepedet. Similarly (δ 0, δ 1,..., δ 1 ) i F{0, 1,..., 1 }. 2
Reducig spaig lists to liearly idepedet oes Lemma 5 (Lemma 2.4 + more). Let V = Spa(v 1,..., v ). The, (v 1,..., v ) is liearly idepedet if ad oly if, for all i, Spa(v 1,..., v i 1, v i+1,..., v ) Spa(v 1,..., v ). (1.1) I other words, a spaig list is liearly idepedet if ad oly if it is miimal. Examples: (1, 0), (0, 1) is a miimal spaig list, hece liearly idepedet. Same is true for (1, 0), (1, 1), or ay spaig list of two vectors i R 2. If the zero vector is i a spaig list, it is ot miimal (hece liearly depedet). Defiitio 6. A liearly idepedet spaig set is called a basis. By the lemma, every spaig set ca be reduced to a basis (Theorem 2.10). Just discard vectors util (1.1) is true! Maximal liearly idepedet lists Lemma 7 (Part of Theorem 2.12). A liearly idepedet list (v 1,..., v ) of vectors of V spas V if ad oly if it is maximal: (v 1,..., v, v) is ot liearly idepedet for all v V. Examples: (1, 0, 0), (1, 1, 0) is liearly idepedet, but ot maximal. (1, 1), ( 2, 1) is a maximal liearly idepedet list, hece it spas. Ay liearly idepedet list which does ot spa ca be ex- Cosequece: teded. The fudametal iequality ad bases Theorem 8 (Theorem 2.6). The legth of every liearly idepedet list is less tha or equal to the legth of every spaig list. Corollary 9 (Theorem 2.14). Every basis has the same legth. Proof. Let (v 1,..., v m ) ad (w 1,..., w ) be bases. Sice (v 1,..., v m ) is liearly idepedet ad (w 1,..., w ) spas, m. Sice (v 1,..., v m ) spas ad (w 1,..., w ) is liearly idepedet, m. 3
Bases, cotiued Corollary 10 (Corollary 2.11 + more). Every fiite-dimesioal vector space V has a basis, ad every basis has legth dim V. Proof. Every spaig list of legth dim V must be miimal, hece liearly idepedet, i.e., a basis. By the previous corollary, all bases have the same legth. Corollary 11 (Theorem 2.12). Every liearly idepedet list i a fiitedimesioal vector space ca be exteded to a basis. Proof. Liearly idepedet lists i V of legth < dim V do t spa, so ca be exteded. Liearly idepedet lists of legth dim V must have legth exactly dim V. Thus, they are maximal ad spa. Examples For R 2, every pair of ozero vectors, either of which is a multiple of the other, is a basis. I F, the delta vectors form a basis. Also ay li. id. list of legth (or ay spaig list of legth ). I F{0, 1,..., 1 }, agai, the delta fuctios form a basis. For F = C, aother basis importat for Fourier trasform: {1, e 2πit, e 4πit,..., e 2( 1)πit, where e iy = cos(y) + i si(y). Lots of bases importat for MP3s, auto-tue, DVDs, etc! The basis theorem Theorem 12 (Propositios 2.8, 2.16, ad 2.17). The followig coditios for a list (v 1,..., v ) i V are equivalet: (a) It is a basis. (b) It is liearly idepedet of legth dim V. (c) It is spaig of legth dim V. (d) For all v V, there exist uique λ 1,..., λ such that v = λ 1 v 1 + +λ v. Proof. By the corollary, (a) implies (b) ad (c). (b) implies (a) sice liearly idepedet sets of legth dim V are maximal. Similarly, (c) implies (a) sice spaig sets of legth dim V are miimal. (a) implies (d): There exist λ 1,..., λ sice v 1,..., v spa. uique by liear idepedece. They are 4
Dimesio Theorem 13 (Propositios 2.7 ad 2.15). If V is fiite-dimesioal ad U V is a subspace, the U is fiite-dimesioal, ad dim U dim V. Proof. Ay liearly idepedet list i U is also liearly idepedet i V, so has legth at most dim V. So, there is a maximal such list, which must be a basis. Sice it has legth at most dim V, we obtai that dim U dim V. 5