3.3 Operations With Vectors, Linear Combinations

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1 Operations With Vectors, Linear Combinations Performance Criteria: (d) Mltiply ectors by scalars and add ectors, algebraically Find linear combinations of ectors algebraically (e) Illstrate the parallelogram method and tip-to-tail method for finding a linear combination of two ectors (f) Find a linear combination of ectors eqalling a gien ector In the preios section a ector x x,x,,x n in n dimensions was starting to look sspiciosly like an n-tple (x,x,,x n ) and we established a correspondence between any point and the position ector with its tip at that point One might wonder why we bother then with ectors at all! The reason is that we can perform algebraic operations with ectors that make sense physically, and sch operations make no sense with n-tples The two most basic things we can do with ectors are add two of them or mltiply one by a scalar, and both are done component-wise: Definition : Addition and Scalar Mltiplication of Vectors Let,,, n and,,,n n, and let c be a scalar Then we define the ectors + and c by + + and c c n n + + n + n n c c c n Note that reslt of adding two ectors or mltiplying a ector by a scalar is also a ector It clearly follows from these that we can get sbtraction of ectors by first mltiplying the second ector by the scalar, then adding the ectors With jst a little thoght yo will recognize that this is the same as jst sbtracting the corresponding components Example (a): For and 9, find +, and + 9 +( ) () ( ) () 9 ( ) 9 9 Addition of ectors can be thoght of geometrically in two ways, both of which are sefl The first way is what we will call the tip-to-tail method, and the second method is called the parallelogram method Yo shold become ery familiar with both of these methods, as they each hae their adantages; they are illstrated below

2 Example (b): Add the two ectors and shown below and to the left, first by the tip-to-tail method, and second by the parallelogram method To add sing the tip-to-tail method moe the second ector so that its tail is at the tip of the first (Be sre that its length and direction remain the same!) The ector + goes from the tail of to the tip of See in the middle below + tip-to-tail method parallelogram method + To add sing the parallelogram method, pt the ectors and together at their tails (again being sre to presere their lengths and directions) Draw a dashed line from the tip of, parallel to, and draw another dashed line from the tip of, parallel to + goes from the tails of and to the point where the two dashed lines cross See to the right aboe The reason for the name of this method is that the two ectors and the dashed lines create a parallelogram Each of these two methods has a natral physical interpretation For the tip-to-tail method, imagine an object that gets displaced by the direction and amont shown by the ector Then sppose that it gets displaced by the direction and amont gien by after that Then the ector + gies the net (total) displacement of the object Now look at that pictre for the parallelogram method, and imagine that there is an object at the tails of the two ectors If we were then to hae two forces acting on the object, one in the direction of and with an amont (magnitde) indicated by the length of, and another with amont and direction indicated by, then + wold represent the net force (In a statics or physics corse yo might call this the resltant force) A ery important concept in linear algebra is that of a linear combination Let me say it again: Linear combinations are one of the most important concepts in linear algebra! Yo need to recognize them when yo see them and learn how to create them They will be central to almost eerything that we will do from here on A linear combination of a set of ectors,,, n (note that the sbscripts now distingish different ectors, not the components of a single ector) is obtained when each of the ectors is mltiplied by a scalar, and the reslting ectors are added p So if c,c,,c n are the scalars that,,, n are mltiplied by, the reslting linear combination is the single ector gien by c +c +c + +c n n Emphasizing again the importance of this concept, let s proide a slightly more concise and formal definition: Definition : Linear Combination A linear combination of the ectors,,, n, all in R n, is any ector of the form c +c +c + +c n n, where c,c,,c n are scalars

3 Note that when we create a linear combination of a set of ectors we are doing irtally eerything possible algebraically with those ectors, which is jst addition and scalar mltiplication! Yo hae seen this idea before; eery polynomial like x x + x is a linear combination of,x,x,x, Those of yo who hae had a differential eqations class hae seen things like ds dy dt +dy dt +y, which is a linear combination of the second, first and zeroth deriaties of a fnction y y(t) Here is why linear combinations are so important: In many applications we seek to hae a basic set of objects (ectors) from which all other objects can be bilt as linear combinations of objects from or basic set A large part of or stdy will be centered arond this idea This may not make any sense to yo now, bt hopeflly it will by the end of the corse Example (c): Fortheectors, 9 and, gie the linear combination + as one ector + 9 Example (d): For the same ectors, and as in the preios exercise and scalars c, c and c, gie the linear combination c +c +c as one ector c +c +c c c c c c 9 c 9c c c c +c c +9c +c c +c +c c c c c Note that the final reslt is a single ector with three components that look sspiciosly like the left sides of a system of three eqations in three nknowns! In the preios two examples we fond linear combinations algebraically; in the next example we find a linear combination geometrically Example (e): In the space below and to the right, sketch the ector for the ectors and shown below and to the left In the center below the linear combination is obtained by the tip-to-tail method, and to the right below it is obtained by the parallelogram method The last example is probably the most important in this section

4 Example (f): Find a linear combination of the ectors ector w and that eqals the We are looking for two scalars c and c sch that c +c w By the method of Example (d) we hae c +c c + c c c c +c c c In the last line aboe we hae two ectors that are eqal It shold be intitiely obios that this can only happen c if the indiidal components of the two ectors are eqal This reslts in the system +c of c c two eqations in the two nknowns c and c Soling, we arrie at c, c It is easily erified that these are correct: We now conclde with an important obseration Sppose that we consider all possible linear combinations of a single ector That is then the set of all ectors of the form c for some scalar c, which is jst all scalar mltiples of At the risk of being redndant, the set of all linear combinations of a single ector is all scalar mltiples of that ector Section Exercises For the two ectors and shown below and to the left, illstrate the tip-to-tail and parallelogram methods for finding + in the spaces indicated Tip-to-tail: Parallelogram: For the ectors,, linear combination + + as one ector and, gie the For the ectors c +c as one ector and, gie the linear combination

5 Gie a linear combination of, that yor answer is correct by filling in the blanks: and w that eqals 9 Demonstrate 9 For each of the following, find a linear combination of the ectors, and that eqals Conclde by giing the actal linear combination, not jst some scalars (a),,, (b),,, (a) Consider the ectors,,, and w If possible, find scalars a, a and a sch that a +a +a w (b) Consider the ectors,,, and w If possible, find scalars b, b and b sch that b +b +b w (c) To do each of parts (a) and (b) yo shold hae soled a system of eqations Let A be the coefficient matrix for the system in (a) and let B be the coefficient matrix for the system in part (b) Use yor calclator to find det(a) and det(b), the determinants of matrices A and B Yo will probably find the command for the determinant in the same men as rref