4 Inner Product Spaces
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1 11.MH1 LINEAR ALGEBRA Summary Notes 4 Ier Product Spaces Ier product s the abstracto to geeral vector spaces of the famlar dea of the scalar product of two vectors or 3. I what follows, keep these key examples md. A er product o a vector space V s a fucto V V, usually deoted u v u v, satsfyg four axoms: 1. u v v u. u v w u v u w 3. u v u v 4. v v 0, wth v v 0 f ad oly f v 0. Defto A vector space V equpped wth a er product space. s called a er product Examples There are may examples of er products o vector spaces. The stadard example s the eucldea er product u v u t v o M 1. Aother class of examples are gve by tegrals, where the vector space V cossts of sutable real-valued fuctos. For example, f V s the space of cotuous fuctos 0 1, the f g s a er product o V. 0 1 f x g x dx A mportat property of a er product s the: Cauchy-Schwartz Iequalty u v u u v v u v V Proof. If u 0 the both sdes are 0 ad there s othg to prove. Suppose the that u 0. Sce 0 u v u v u u u v v v the theory of quadratc equatos tells us that u v 4 u u v v 0
2 As the famlar examples of ad 3, we ca use the er product to defe the legth, or orm of a vector as v v v ad the agle formed (at the org) by u ad v as cos u v u (Ths makes sese, sce the Cauchy-Schwartz equalty mples that the rght had sde les betwee 1 ad 1.) I partcular, u v are orthogoal f they make a agle, e f u v 0. A set of vectors v 1 v s orthogoal f v v j 0 wheever j. If addto each v has orm 1, e v v 1, the t s orthoormal. If B v 1 v s a orthogoal bass for V, the the scalars 1 that appear the expresso of a vector v as a lear combato v 1v 1 v (the coordates of v wth respect to B) ca easly be computed as v v v v If, moreover, B s orthoormal (e v v 1 for all ), the ths smplfes to: v v To form a orthoormal bass from a exstg bass u, we use: Gram-Schmdt Process 1. Set v 1. Assume that we have costructed a orthoormal bass v 1 v k for Spa u k, where 0 k. Defe u k 1 u k 1 k 1 u k 1 v v Note that u k 1 0, sce u k 1 Spa v 1 v k. Set v k 1 u k 1 u k 1
3 3. REPEAT UNTIL k. Example Use the Gram-Schmdt process to trasform to a orthoormal bass for. 3 4,, so u 6, so u v 1 39, ad Fally 3 v 1 u u 39 v 1 u v 4 u 8 Extedg orthoormal bases for subspaces Suppose we have formed a orthoormal bass u 1 for a hyperplae H x 1 x 1x 1 x. There s a quck way to exted ths to a orhtoormal bass for : let v be the ormal vector 1 t to H. (The u v 0.) Set u v v The s the desred bass for. More geerally, f U s the soluto space of a system of homogeous equatos AX 0, the the colum space of A t s orthogoal to U, so f B 1 B are orthoormal bases for U ad for the colum space of A respectvely, the B 1 B s a orhtoormal bass for. Coordates Suppose that V s a fte dmesoal vector space, ad B v 1 v s a bass for V. That meas that every elemet v V ca be uquely expressed the form u v 1v 1 v 3
4 wth. The are called the coordates of v wth respect to the bass B, ad the colum matrx X 1. M 1 s called the coordate matrx or the coordate vector of v wth respect to B. If V s a er product space, ad B s a orthogoal bass, the the coordates ca be readly calculated as: v v v v If fact B s orthoormal, the the deomator ths expresso s 1, ad we get the smpler expresso v v Chage of bass Suppose that B v 1 v ad B v 1 v are two dfferet bases for V. How do the coordates of a elemet v V wth respect to B relate to the coordates of v wth respect to B? Suppose X 1. s the coordate matrx of v wth respect to B, ad Y 1. s the coordate matrx of v wth respect to B. Sce v 1v 1 v t follows that Y 1C 1 C where C s the coordate matrx of v wth respect to B. If A s the colums C 1 C, the Y AX matrx wth The matrx A s called the trasto matrx from B to B. Some obvous propertes of trasto matrces: 4
5 1. The trasto matrx from B to B s the detty matrx I, for ay bass B.. If A s the trasto matrx from B to B, ad A s the trasto matrx from B to a thrd bass B, the the matrx product A A s the trasto matrx from B to B. (To see ths, suppose X Y Z are the coordate matrces of some elemet v V wth respect to the three bases B B B. The Y AX ad Z A Y, so Z A A X.) 3. Puttg together the two propertes above, we see that, f A s the trasto matrx from B to B, the A s vertble, ad ts verse A 1 s the trasto matrx from B to B. Theorem Let V be a er product space ad let B B be two orthoormal bases for V. The the trasto matrx from B to B s orthogoal, that s A t A 1. Proof. By defto, the j -etry of A s the v -coordate of v j, whch s v v j, sce B s orthoormal. But B s also orthoormal, so by the same argumet ths s also the j -etry of the trasto matrx from B to B, e of A 1. Hece A 1 A t, as clamed.
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