an introduction to linear algebra by v krishnamurthy

... b2 v1 + (sv1 + tv2 ) ∈ Span {v1 , sv1 + tv2 } t t Thus, Span {v1 , v2 } ⊆ Span {v1 , sv1 + tv2 } Hence Span {v1 , v2 } = Span {v1 , sv1 + tv2 } D8 (a) TRUE We can rearrange the equation to get −tv1 ... D3 We have v1 + v2 = (v1 + v2 ) · (v1 + v2 ) = v1 · v1 + v1 · v2 + v2 · v1 + v2 · v2 = v1 + + + v2 = v1 + v2 Copyright c 2013 Pearson Canada Inc ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 24 Chapter Euclidean Vector Spaces ... Hence, Span {v1 , sv1 + tv2 } ⊆ Span {v1 , v2 } Since t we get that v2 = −st v1 + 1t (sv1 + tv2 ) Hence, if v ∈ Span {v1 , v2 }, then v = b1 v1 + b2 v2 = b1 v1 + b2 = b1 − −s v1 + (sv1 + tv2 ) t t

Ngày tải lên: 20/08/2020, 13:34

... An Introduction to LINEAR ALGEBRA An Introduction to LINEAR ALGEBRA Ravi P Agarwal and Cristina Flaut CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, ... Paiu, and Maria Paiu Contents Preface Linear Vector Spaces ix Matrices 11 Determinants 23 Invertible Matrices 31 Linear Systems 43 Linear Systems (Cont’d) 51 Factorization 61 Linear Dependence and ... segment linear combination linear functional linear mapping linear operator linear transformation linearly dependent set of vectors 150 24 62 198 183 91 123 145 104 127 97 97 97 67, 69 230 Index linearly

Ngày tải lên: 15/09/2020, 15:45

... avvT )T = I T − a(vvT )T = I − a (v T T v T ) = I − avvT = A AA = (I − avvT )(I − avvT ) = I − 2avvT +a2 (vvT )(vvT ) = I − 2avvT +a[2/vvT ](vvT vvT ) = I − 2avvT +(2avvT ) = I 57 Ax = I x −a(vvT ... combination of v1 and v2 for any value of a 48 A nontrivial solution is: 1v1 + 0v2 + 0v3 = θ 49 = θ T θ = (a1 v1 + a2 v2 + a3 v3 )T (a1 v1 + a2 v2 + a3 v3 ) = a21 v3 , so = 0, i = 1, 2, v1 +a22 v2 +a23 ... a1 v1 + a2 v2 + a3 v3 = θ, where some = 0, then a1 v1 + a2 v2 + a3 v3 + 0v4 = θ 51 If θ = a1 v1 + a2 (v1 + v2 ) + a3 (v1 + v2 + v3 ) then θ = (a1 + a2 + a3 )v1 + (a2 + a3 )v2 + a3 v3 Since {v1

Ngày tải lên: 20/08/2020, 12:03

... Lie algebras and other non- associative algebras which arise through such mechanisms as the deriva- tion algebra. Let A be any algebra over F . By a derivation of A is meant a linear operator ... ring R to be an additive abelian group with a second law of composition, multiplication, which satisfies the distributive laws (2). We define an algebra A over a field F to be a vector space over F ... associative algebras do not involve associativity in any way, and so may fruitfully be employed in the study of nonassociative algebras. For example, we say that two algebras A and A  over F are

Ngày tải lên: 28/06/2014, 19:20

... we will try to identify how many columns are on the site. We do this by using the ORDER BY command. The command can be used in a few variations listed below. CHAPTER 5 – How many columns? ... SQL 4 and 5 Ok we have covered how to get the version of the SQL server now we will move on with the next steps we use to gain more information We will now use information schema to find ... below As you can see it is not happy with the syntax ‘tbl_accounts’ If you get an error like this it means you have to encode the table name into hex format to do this you can visit this website

Ngày tải lên: 12/07/2014, 19:20

... statement and the contrapositive are logically equivalent It is also true that the converse and the inverse are equivalent to each other But neither the converse nor the inverse are equivalent to the ... way to prove the biconditional p ⇔ q is to prove the conditional p ⇒ q and its converse q ⇒ p   (c) ( p ⇒ q) ⇔ (~ q ⇒ ~ p) One way to prove the conditional p ⇒ q is to prove its contrapositive ... proved the general statement “" n,  q  (n),” we can apply it to any particular case For example, we know that g(124, 125) = 125 without having to any computation This is an example of deductive

Ngày tải lên: 14/08/2017, 16:16

... apply to all the integers Copyright © 2013, 2005, 2001 Pearson Education, Inc Section 1.4, Slide 1-4 How we prove quantified statements? To prove an existential statement ∃x p (x), we have to prove ... contrapositive implication, ~ q ⇒ ~ p, can be written as If for every x in [0, 1], f (x) = 0, This is much easier to prove ∫0 f ( x) dx = then Instead of having to conclude the existence of an x in ... implies q, we can so by showing that p and not q leads to a contradiction In either case the contradiction can involve part of the hypothesis or some other statement that is known to be true Copyright

Ngày tải lên: 14/08/2017, 16:16

... given x in A can correspond to only one y in B This means that f  –  1 is injective, and hence bijective When f is followed by f  That is, ( f  –  1  ○ –  1, the effect is to map x in A onto ... injective g ○ f  is surjective (c) If f and g are bijective, then g ○ f  is bijective Inverse Functions Given a function f : A → B, we have seen how f determines a relationship between That is, given ... ○ f is bijective, so g ○ f has an inverse denoted by ( g ○ f ) − 1, and this inverse maps C onto A g  f ○ f g A B f  –1 C –1 g  ( g  f ) ○   –1 –1 –1 = f    g  ○ We are asked to verify the equality

Ngày tải lên: 14/08/2017, 16:16

... 2.2.7 Let A and B be sets A relation between A and B is any subset R of A × B We say that an element a in A is related by R to an element b in B if (a, b) ∈ R, and we often denote this by writing ... the plane and let R be the relation “is parallel to. ” It is reflexive (if we agree that a line is parallel to itself), symmetric, and transitive (c) Let S be the set of all people who live in ... y R z, then x R z (transitive property) Example 2.2.10 Determine which properties apply to each relation (a) Define a relation R on by x R y if x ≤ y It is reflexive and transitive, but not symmetric

Ngày tải lên: 14/08/2017, 16:16

... bijective function from S onto T It is easy to see that “~” is an equivalence relation on any family of sets, and it partitions the family into disjoint equivalence classes With each equivalence ... odd if n is even We can see this as follows: Section 2.4, Slide 1-8 Given surjections f : → S and g : → T, and the even integers with g to count T let h use the odd integers with f to count S h ... corresponds to our intuitive feeling about the relative sizes of subsets (b) This is the reflexive property (c) This is the transitive property (d) This means that the order of m and n as integers

Ngày tải lên: 14/08/2017, 16:16

... polynomials, it is not difficult to verify that is a field quotient such as above is positive iff We can define an order on by saying that a an and bk have the same sign; that is, an? ?⋅ bk > 3x + x − > 0, ... We have Thus (–1) ⋅ x by M2 (commutative) = x + x ⋅ (–1) = x ⋅ (1) + x ⋅ (–1) by M4 (mult identity) = x ⋅ [1 + (–1)] by DL (distributive law) = x⋅0 by A5 (add inverse) = by part (b) = – x by the ... (– z) by A5 (add inverse) and A1 (addition) x + [z + (– z)] = y + [z + (– z)] x+0 = y+0 x = y Copyright © 2013, 2005, 2001 Pearson Education, Inc by A3 (assoc property) by A5 (add inverse) by A4

Ngày tải lên: 14/08/2017, 16:16

... important to realize that the two parts are separate arguments; we start over from scratch in proving the converse and can use nothing that was derived about the point x in the first part By using ... included and a round parenthesis if the endpoint is not included The set [a, b] is called a closed interval and the set (a, b) is called an open interval We also have occasion to refer to the unbounded ... so there is an x in S to choose This might seem to be an unwarranted assumption, but really it is not If S is the empty set, then of course S ⊆ T, so the only nontrivial case to prove is when

Ngày tải lên: 14/08/2017, 16:16

... suppose that x > : r > and r < p} and x = sup S • • ( x− ) p m p • x We want to find something positive, say 1/m, that we can subtract from x and still have its square be greater than p This will contradict ... 10 upper bounds Then S is bounded above by 8, 9, 8.5, π , and any other real number greater than or equal to 8.Since ∈ S, we have max S = Similarly, S has many lower bounds, including 2, which ... 1, ∈ S and S is nonempty Thus S is bounded above by p, Let x = sup S and by the completeness axiom, sup S exists as a real number It is clear that x > 0, and we claim that x  = p To prove this,

Ngày tải lên: 14/08/2017, 16:16

... called an open cover of S If G ⊆ F and G is also an open cover of S, then G is called a subcover of S Thus S is compact iff every open cover of S contains a finite subcover Example 3.5.2 To see ... closed and Choose a member K of F and suppose that no point of K belongs to every Kα Then every point of K belongs to some Fα That is, the sets Fα form an open cover ofSince K K is compact, there ... many indices a contradiction   {Kα : α ∈ A } ≠ ∅ ♦ Section 3.5, Slide 12 Corollary 3.5.8 The Nested Intervals Theorem Let F = {An? ?: n ∈  } be a family of closed bounded intervals in An? ?+1 ⊆ An

Ngày tải lên: 14/08/2017, 16:16

... intersections and unions relate to open sets and closed sets? Theorem 3.4.10 (a) The union of any collection of open sets is an open set (b) The intersection of any finite collection of open sets is an ... Section 3.4 Topology of the Real Numbers Copyright © 2013, 2005, 2001 Pearson Education, Inc Section 3.4, Slide Recall that the distance between two real numbers x and y is given by | x – y | Many of ...  1/n, ∞ Each of these sets is an open set (an open interval)  1/n) A ? n =1 n I ∞ A n =1 n = {0}, which is not open So the open sets have “shrunk down” (by intersecting) to a set that is not open

Ngày tải lên: 14/08/2017, 16:16

... vectors. Any linearly independent set of vectors can be extended to a basis by adding (linearly independent) vectors so that the set spans the space. Likewise, any spanning set of vectors can ... orthogonal vectors. Recall that any two vectors v and w are orthogonal if their inner product is zero. Two subspaces V and W can also be orthogonal, provided that every vector v in V is orthogonal to ... X can be factored into X = USV T (70) where U is orthogonal and m by m, V is orthogonal and n by n, and S is m by n and diagonal. The non-zero entries of S are referred to as the singular values...

Ngày tải lên: 17/01/2014, 04:20

... Bruce Worthington and Greg Ganger at the University of Michigan took this approach and managed to fine-tune the controller-overhead and bus-transfer components of a model similar to ours. They achieved ... Failure to model disk drive behavior can result in quantitative—and in extreme cases, qualitative—errors in an analysis. Careful modeling is neither too difficult nor too costly. We have provided ... the behavior of a real drive. Related work. Disk drive models have been used ever since disk drives became available as storage devices. Because of their nonlinear, state-dependent behavior, disk...

Ngày tải lên: 12/09/2012, 14:16

... it to a file  The value of the stream variable can be thought of as the file it is connected to  Can have its value changed  Changing a stream value means disconnecting from one file and ... Files?  Files allow you to store data permanently!  Data output to a file lasts after the program ends  An input file can be used over and over  No typing of data again and again for testing  Create ... output  Input is delivered to your program via a stream object  Input can be from  The keyboard  A file  Output is delivered to the output device via a stream object  Output can be to  The screen  A...

Ngày tải lên: 12/09/2012, 22:49

... systematic and organised approach to their work and use appropriate tools and techniques depending on the problem to be solved, the development constraints and the resources available. ©Ian Sommerville ... what the customer wants ã Evolution - changing the software in response to changing demands. ©Ian Sommerville 2004 Software Engineering, 7th edition. Chapter 1 Slide 1 An Introduction to Software Engineering âIan ... efficiency and usability.  The software process consists of activities that are involved in developing software products. Basic activities are software specification, development, validation and evolution. ...

Ngày tải lên: 14/09/2012, 11:26

... that many trivial, or relatively trivial, externalities are ignored. However, what may lead to a trivial cost for each individual affected may in aggregate involve non-trivial and even substantial ... regulatory enforcement is the width of discretion within regulatory systems (in the hands of both public and private actors), providing ample scope for human action, error, manipulation and creativity. 10 ... introduction to the subject. Morgan and Yeung set out a diverse and stimulating selection of materials and give them context with a compre- hensive and critical commentary. By adopting an interdisciplinary...

Ngày tải lên: 21/09/2012, 10:39