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# Introduction to Abstract Algebra MATH 113

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This 2 page Class Notes was uploaded by Kavon Feest on Thursday October 22, 2015. The Class Notes belongs to MATH 113 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/226604/math-113-university-of-california-berkeley in Mathematics (M) at University of California - Berkeley.

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Date Created: 10/22/15

Math 110 Notes for the lecture on February 10 2005 The first part of the lecture will correspond to the end of the notes that were posted for February 8 I will discuss change of basis For details on this see the notes for February 8 The situation there is that we have a linear T V a W where V and W are finite dimensional We assume that V has bases and and that analogously W has bases y and M Imagine that we can write 5 in terms of Q and y in terms of 7 Then there are three obvious matrices lying around A Q 1Vlgn R Then the formula to remember is that my AQ As I said details are in the last set of notes A very important example is that where W V y and y 5 The book writes Tl3 for and employs analogous notation for Tl6 We then get the formula Tl3 RillTlBR which is very important The formula states that Tly is the conjugate of Tl3 by R the word conjugate is undoubtedly familiar to you if you7ve taken Math 113 For an example take F to be the field of complex numbers and let V W F2 Let a Tb Let b a 39 T LA Then of course Tl3 A Let 5 be the alternative basis 1 7i l Then we should be able to check in class that Tly is the 2 gtlt 2 diagonal matrix whose diagonal Q be the standard basis of V Let a and b be complex numbers let A entries are a bi and a 7 bi The matrix R here is i Just to answer someone7s question we are not going to discuss 27 We will however discuss 26 This section concerns the all important topic of dual spaces If V is an F vector space its dual Vquot is the space V F of linear transformations V a F Such linear transformations are called linear functionals If V is finite dimensional and has ordered basis 111 v then there are n different elements of Vquot staring us in the face These are the coordinate functions f1 fn that are defined by the basis Namely for v E V we may write uniquely v 2211 aim with ai E F The functional f1 maps 1 to the coef cient ai It is easy to see that fi is linear lt satisfies the key formula fivj 61 where Flj is the Kronecker delta function which by definition is 1 when i j and 0 when i andj are distinct For f E Vquot we see that f is determined by the numbers fvZ for i l n This follows from a general theorem which states that a linear map T V a W is determined by the vectors Tvi in W Explicitly here if v Z 1in as before then H Z aifvi39 11 Once we know the fvZ we have a recipe for nding fanything write each vector of V in terms of the basis vectors and use the formula that7s displayed just above As the book points out V has dimension 11 when V has dimension 11 Indeed V W is known by us to have dimension nm when V has dimension n and W has dimension m Here W is the l dimensional space F F1 A more precise result is that the fi form a basis of V This basis depends on Q 111 vn of course and is said to be the basis of Vquot that is dual to the basis Q The basis dual to Q is denoted 5 Let7s prove this result First we should check that the fi are linearly dependent Suppose that we have cifi 0 Then by definition we have that GHQU 0 for all 1 E V If we put 1 v where j is a number between 1 and n then the sum collapses to cj because of the Kronecker delta business Thus we have cj 0 for each j thus the vanishing linear combination c was the trivial linear combination with all coef cients 0 and we have established the required linear independence Now lets show that each f E Vquot is a linear combination of the fi Let f be given and set c fvZ for each i Then the claim is that f cifi To see this we need to show that the difference between the two sides of the equation which is an element of Vquot vanishes on every 1 E V This forces the difference to be the 0 element of V Said differently we want the null space of the difference to be all of V However the Kronecker delta business shows that the null space of the difference contains each of the basis vectors vj Since the null space is a subspace of V and since the basis vectors span V the null space of the difference is indeed the entire vector space The next topic concerns duals of linear maps T V a W where V and W are finite dimensional There7s a natural map W a Vquot which is composition with T g E W gt gt gT goT E V This map is called T by most authors and Tt be our authors We7ll call it Tt The lower case 15 means transpose The reason for this terminology becomes clear if V is given with a basis and W with a basis 7 Then we have two matrices at hand T and Note that the first is an m gtlt 11 matrix while the second is an n gtlt m matrix The fundamental result is that the two matrices are transposes of each other This result is proved by direct computation cf p 121 of our text PM do the computation at the board keep those pencils sharp

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