(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:

Transcription

1 Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts, which track how elemetary row operatios affect determiats (a) Swappig ay two rows chages the sig of the determiat proof This is clear for matrices, sice a b c d c d ad bc, a b cb ad For 3 3 determiats, expad across the row ot beig swapped, ad use the swap property to deduce the result Prove the geeral result by iductio oce it's true for matrices you ca prove it for ay matrix, by expadig across a row that was't swapped, ad applyig the result (b) Thus, if two rows i a matrix are the same, the determiat of the matrix must be zero o the oe had, swappig those two rows leaves the matrix ad its determiat uchaged; o the other had, by (a) the determiat chages its sig The oly way this is possible is if the determiat is zero (a) If you factor a costat out of a row, the you factor the same costat out of the determiat Precisely, usig i ith row of A, ad writig * i i i c i * c i * proof expad across the i th row, otig that the correspodig cofactors do't chage, sice they're computed by deletig the i th row to get the correspodig miors det A j a i j C i j j * c a Ci i j j c j * a Ci i j j c det A * (b) Combiig (a) with (b), we see that if oe row i A is a scalar multiple of aother, the det A (3) If you replace row i of A by its sum with a multiple of aother row, the the determiat is uchaged! Expad across the i th row

2 i c k k j a c a C a C i j k j i j j i j i j c a C det A j k j i j c k k det A Remark The aalogous properties hold for correspodig "elemetary colum operatios" I fact, the proofs are almost idetical, except you use colum expasios

3 Exercise ) Recompute 3 from yesterday (usig row ad colum expasios we always got a aswer of 5 the) This time use elemetary row operatios (ad/or elemetary colum operatios) Exercise ) Compute Maple check > with LiearAlgebra > A Matrix 4, 4,,,,,,,,,,,,,,,, ; Determiat A ; A ()

4 Theorem Let A The A exists if ad oly if det A proof We already kow that A exists if ad oly if the reduced row echelo form of A is the idetity matrix Now, cosider reducig A to its reduced row echelo form, ad keep track of how the determiats of the correspodig matrices chage As we do elemetary row operatios, if we swap rows, the sig of the determiat switches if we factor o-zero factors out of rows, we factor the same factors out of the determiats if we replace a row by its sum with a multiple of aother row, the determiat is uchaged Thus, A c A c A c c N rref A where the ozero c k 's arise from the three types of elemetary row operatios If rref A I its determiat is, ad A c c N If rref A I the its bottom row is all zeroes ad its determiat is zero, so A c c N Thus A A exists if ad oly if rref A I if ad oly if Remark Usig the same ideas as above, you ca show that det A B det A det B This is a importat idetity that gets used, for example, i multivariable chage of variables formulas for itegratio, usig the Jacobia matrix (It is ot true that det A B det A det B ) Here's how to show det A B det A det B The key poit is that if you do a elemetary row operatio to AB, that's the same as doig the elemetary row operatio to A, ad the multiplyig by B With that i mid, if you do exactly the same elemetary row operatios as you did for A i the theorem above, you get A B c A B c A B c c N rref A B If rref A I, the from the theorem above, A c c N, ad we deduce A B A B If rref A I, the its bottom row is zeroes, ad so is the bottom row of rref A B Thus A B ad also A B

5 There is a "magic" formula for the iverse of square matrices A (called the "adjoit formula") that uses the determiat of A alog with the cofactor matrix of A I order to uderstad the magic formula for matrix iverses, we first eed to talk about matrix trasposes Defiitio Let B m b i j The the traspose of B, deoted by B T is a m matrix defied by etry i j B T etry j i B b j i The effect of this defiitio is to tur the colums of B ito the rows of B T etry i col j B b i j etry i row j B T etry j i B T b i j Ad to tur the rows of B ito the colums of B T etry j row i B b i j etry j col i B T etry j i B T b i j Exercise 3) explore these properties with the idetity T

6 Theorem Let A, ad deote its cofactor matrix by cof A C i j, with C i j i j M i j, ad M i j the determiat of the matrix obtaied by deletig row i ad colum j from A Defie the adjoit matrix to be the traspose of the cofactor matrix Adj A cof A T The, whe A exists it is give by the formula A det A Adj A Exercise 4) Show that i the case this reproduces the formula a b d b c d ad bc c a Exercise 5) For our fried A 3 we worked out cof A ad det A 5 Use the Theorem to fid A ad check your work Does the matrix multiplicatio relate to the dot products we computed betwee various rows of A ad rows of cof A?

7 Exercise 6) Cotiuig with our example, 5 6 A 3 cof A Adj A a) The, etry of A Adj A is Explai why this is det A, expaded across the first row 6b) The, etry of A Adj A is Notice that you're usig the same cofactors as i (4a) What matrix, which is obtaied from A by keepig two of the rows, but replacig a third oe with oe of those two, is this the determiat of? 6c) The 3, etry of A Adj A is 3 6 What matrix (which uses two rows of A) is this the determiat of? If you completely uderstad 6abc, the you have realized why A Adj A det A I for every square matrix, ad so also why A det A Adj A Precisely, etry i i A Adj A row i A col i Adj A row i A row i cof A det A, expaded across the i th row O the other had, for i k, etry k i A Adj A row k A col i Adj A row k A row i cof A This last dot produce is zero because it is the determiat of a matrix made from A by replacig the i th row with the k th row, expadig across the i th row, ad wheever two rows are equal, the determiat of a matrix is zero

8 There's a related formula for solvig for idividual compoets of x whe A x b has a uique solutio ( x A b ) This ca be useful if you oly eed oe or two compoets of the solutio vector, rather tha all of it Cramer's Rule Let x solve A x b, for ivertible A The x k det A k det A where A k is the matrix obtaied from A by replacig the k th colum with b proof Sice x A b the k th compoet is give by x k etry k A b etry k A Adj A b Notice that col k cof A A row k Adj A b A col k cof A b b is the determiat of the matrix obtaied from A by replacig the k th colum by b, where we've computed that determiat by expadig dow the k th colum! This proves the result 5 x Exercise 7) Solve 7 4 y 7a) With Cramer's rule 7b) With A, usig the adjoit formula

9 Math 5-4 Week 7 otes Sectios 4-43 vector space cocepts Tues Feb Fiish sectio 36 o Determiats ad coectios to matrix iverses Use last week's otes The if we have time o Tuesday, begi 4-43 The vector space m ad its subspaces; cocepts related to liear combiatios of vectors We ever wrote it dow carefully i Chapter 3, but for ay atural umber m,, 3 the space m may be thought of i two equivalet ways I both cases, m cosists of all possible m tuples of umbers (i) We ca thik of those m case we ca write (ii) We ca thik of those m case we ca write tuples as represetig poits, as we're used to doig for m,, 3 I this m x, x,, x m, st x, x,, x m tuples as represetig vectors that we ca add ad scalar multiply I this x m x x m, st x, x,, x m Sice algebraic vectors (as above) ca be used to measure geometric displacemet, oe ca idetify the two models of m as sets by idetifyig each poit x, x,x m i the first model with the displacemet vector T x x, x,x m from the origi to that poit, i the secod model, ie the positio vector (Notice we just used a traspose, writig a colum vector as a traspose of a row vector) Oe of the key themes of Chapter 4 is the idea of liear combiatios These have a algebraic defiitio (that we've see before i Chapter 3 ad repeat here), as well as a geometric iterpretatio as combiatios of displacemets, as we will review i our first few exercises Defiitio If we have a collectio of vectors v, v i m, the ay vector v m that ca be expressed as a sum of scalar multiples of these vectors is called a liear combiatio of them I other words, if we ca write v c v v c v, the v is a liear combiatio of v, v The scalars c,,, c are called the liear combiatio coefficiets

10 Example You've probably see liear combiatios i previous math/physics classes For example you might have expressed the positio vector r as a liear combiatio r x i y j z k where i, j, k represet the uit displacemets i the x,y,z directios, respectively Sice we ca express these displacemets usig Math 5 otatio as i, j, k we have x x i y j z k x y z y z Remarks Whe we had free parameters i our explicit solutios to liear systems of equatios A x b back i Chapter 3, we sometimes rewrote the explicit solutios usig liear combiatios, where the scalars were the free parameters (which we ofte labeled with letters that were t, t 4, t 3 etc, rather tha with "c's") Whe we retur to differetial equatios i Chapter 5 -studyig higher order differetial equatios - the the explicit solutios will also be expressed usig "liear combiatios", just as we did i Chapters -, where we used the letter "C" for the sigle free parameter i first order differetial equatio solutios Defiitio If we have a collectio y, y,, y of fuctios y x defied o a commo iterval I, the ay fuctio that ca be expressed as a sum of scalar multiples of these fuctios is called a liear combiatio of them I other words, if we ca write y c y y c y, the y is a liear combiatio of y, y,, y The reaso that the same words are used to describe what look like two quite differet settigs, is that there is a commo fabric of mathematics (called vector space theory) that uderlies both situatios We shall be explorig these cocepts over the ext several lectures, usig a lot of the matrix algebra theory we've just developed i Chapter 3 This vector space theory will tie i directly to our study of differetial equatios, i Chapter 5 ad subsequet chapters

11 Exercise ) (Liear combiatios i this will also review the geometric meaig of vector additio ad scalar multiplicatio i terms of et displacemets) Ca you get to the poit, 8, from the origi,, by movig oly i the (±) directios of v ad v? Algebraically, this meas we wat to solve the liear combiatio problem 3 c 3 a) Plot the poits, ad, 3, which have positio vectors v, T ad v, 3 T Compute v v, v v Plot the poits for which these are the positio vectors Plot the lie of poits havig positio vectors 8 v t v, t Note Depedig o where you took multivariable calculus you may have writte this parametric lie i various ways x t y 3 t OR x, y t i 3 t j b) Superimpose a grid related to the displacemet vectors v oto the graph paper below, ad, recallig that vector additio yields et displacemet, ad scalar multiplicatio yields scaled displacemet, try to approximately solve the liear combiatio problem above, geometrically c) Rewrite the liear combiatio problem as a matrix equatio, ad solve it exactly, algebraically

12 c) Ca you get to ay poit x, y i, startig at, ad movig oly i directios parallel to v? Argue geometrically ad algebraically How may ways are there to express x, y T as a liear combiatio of v ad v? Defiitio The spa of a collectio of vectors v, v i m is the collectio of all vectors w which ca be expressed as liear combiatios of v, v We deote this collectio as spa v, v Remark The mathematical meaig of the word spa is related to the Eglish meaig - as i "wig spa" or "spa of a bridge", but it's also differet The spa of a collectio of vectors goes o ad o ad does ot "stop" at the vector or associated edpoit Example ) I Exercise, cosider the spa v spa This is the set of all vectors of the form c c with free parameter c This is a lie through the origi of described parametrically, that we're more used to describig with implicit equatio y x (which is short for x, y st y x (More precisely, spa v is the collectio of all positio vectors for that lie ) Example I Exercise we showed that the spa of v ad v 3 is all of

13 Exercise ) Cosider the two vectors v,, T,, T 3 a) Sketch these two vectors as positio vectors i 3, usig the axes below b) What geometric object is spa v? Sketch a portio of this object oto your picture below Remember though, the "spa" cotiues beyod whatever portio you ca draw c) What geometric object is spa v? Sketch a portio of this object oto your picture below Remember though, the "spa" cotiues beyod whatever portio you ca draw

14 d) What implicit equatio must vectors x, y, z T satisfy i order to be i spa v? Hit For what x, y, z T ca you solve the system c z for c,? Write this a augmeted matrix problem ad use row operatios to reduce it, to see whe you get a cosistet system for c, x y

15 Wedesday Feb We've bee talkig about "liear combiatios" of vectors Fiish Tuesday's otes ad the cotiue the discussio here Exercise a) What is the defiitio of "a liear combiatio" of the vectors v, v " b) What is the "spa" of the vectors v, v? Yesterday we iterpreted liear combiatios geometrically Ad, we oticed that to aswer atural questios we eded up usig matrix theory from Chapter 3 This is because Exercise ) By carefully expadig the liear combiatio below, check tha i m, the liear combiatio a a a c a a c a a m a m a m is always just the matrix times vector product a a a c a a a a m a m a m c Thus liear combiatio problems i m ca usually be aswered usig the liear system ad matrix techiques we've just bee studyig i Chapter 3 This will be the mai theme of Chapter 4 We've see this theme i actio, i exercises, i Tuesday's otes

16 Whe we are discussig the spa of a collectio of vectors v, v we would like to kow that we are beig efficiet i describig this collectio, ad ot wastig ay free parameters because of redudacies This has to do with the cocept of "liear idepedece" Defiitio a) The vectors v, v are liearly idepedet if o oe of the vectors is a liear combiatio of (some) of the other vectors The logically equivalet cocise way to say this is that the oly way ca be expressed as a liear combiatio of these vectors, c v v c v, is for all the liear combiatio coefficiets c c b) v, v are liearly depedet if at least oe of these vectors is a liear combiatio of (some) of the other vectors The cocise way to say this is that there is some way to write as a liear combiatio of these vectors c v v c v where ot all of the c j (We call such a equatio a liear depedecy Note that if we have ay such liear depedecy, the ay v j with c j is a liear combiatio of the remaiig v k with k j We say that such a v j is liearly depedet o the remaiig v k ) Note Two o-zero vectors are liearly idepedet precisely whe they are ot multiples of each other For more tha two vectors the situatio is more complicated Example (Refer to Exercise Tuesday) The vectors v, v 3, v 3 8 Tuesday ad as we ca quickly recheck, i are liearly depedet because, as we showed o 35 We ca also write this liear depedecy as (or ay o-zero multiple of that equatio) 35v 5v v 3

17 Exercise 3) Are the vectors v 3 liearly idepedet? How about v, v 3 8?

18 Exercise 4) For liearly idepedet vectors v, v, every vector v i their spa ca be writte as v d v d v d v uiquely, ie for exactly oe choice of liear combiatio coefficiets d, d, d This is ot true if vectors are depedet Explai these facts (You ca illustrate these facts with the vectors i Exercise 3) Exercise 5) (Refer to Exercise i Tuesday's otes) 5a) Are the vectors v, v liearly idepedet?

19 5b) Show that the vectors v, v are liearly depedet (eve though o two of them are scalar multiples of each other) What does this mea geometrically about the spa of these three vectors? Hit You might fid this computatio useful > with LiearAlgebra ReducedRowEcheloForm 6 4 ; 3 () Exercise 6) Are the vectors v, v liearly idepedet? What is their spa? Hit, w 3 > ReducedRowEcheloForm ; ()

20 Math 5-4 Fri Feb Cocepts related to liear combiatios of vectors Exercise ) Vocabulary review (these eed to be memorized!) A liear combiatio of the vectors v, v is The spa of v, v is The vectors v, v are liearly idepedet iff The vectors v, v are liearly depedet iff Keep recallig that for vectors i m all liear combiatio questios ca be reduced to matrix questios because ay liear combiatio like the oe o the left is actually just the matrix product o the right a a a c a a a a a c c a a c a c a c a a a a a m a m a m c a m a m a m a m a m c