8.1 Span of aSet ofVectors - Oregon Institute of Technology

1 Span of a Set of VectorsPerformance Criteria:8. (a) Describe the span of a set of vectors inR2orR3as a line or plane containinga given set of points.(b) Determine whether a vectorwis in the span of a set{v1,v2, ..,vk}of it is, writewas a linear combination ofv1,v2, .., : Thespan of a setSof vectors, denoted span(S) is the set ofall linear combinations of those vectors. Example (a):Describe the span of the setS= 100 , 010 thatANYvector with a zero third component can be written asa linear combination of these two vectors: ab0 =a 100 +b 010 All the vectors withx3= 0(orz= 0) are thexyplane inR3, so thespan of this set is thexyplane. Geometrically we can see the samething in the picture to the right. 010 100 ab0 xyz Example (b):Describe span 1 20 , 310.

2 By definition, the span of this set is all vectorsvof the formv=c1 1 20 +c2 310 ,which, because the two vectors are not scalar multiples of each other, we recognize as being a plane through theorigin. It should be clear that all vectors created by such a linear combination will have a third component of zero,so the particular plane that is the span of the two vectors is thexy-plane. Algebraically we see that any vector [a, b,0]in thexy-plane can be created by a 3b7 1 20 + 2a+b7 310 = a 3b7 2a+6b70 + 6a+3b72a+b70 = 7a77b70 = ab0 You might wonder how one would determine the scalarsa 3b7and2a+b7. You will see how this is done in the exercises!At this point we should make a comment and a couple observations:104 First, some language: we can say that the span of the two vectorsin Example (b) is thexy-plane, but wealso say that the two vectors span thexy-plane.

3 That is, the word span is used as either a noun or a verb,depending on how it is used. Note that in the two examples above we considered two different sets of two vectors, but in each case the spanwas the same. This illustrates thatdifferent sets of vectors can have the same span. Consider also the fact that if we were to include in either of the two sets additional vectors that are also in thexy-plane, it would not change the span. However, if we were to add another vector not in thexy-plane, thespan would increase to all ofR3. In either of the preceding examples, removing either of the two given vectors would reduce the span to a linearcombination of a single vector , which is a line rather than a plane. But insome cases, removing a vector froma set does not change its span.

4 The last two bullet items tell us thatadding or removing vectors from a set of vectors may or may notchangeits is a somewhat undesirable situation that we will remedy in the nextchapter. It may be obvious, but it is worth emphasizing that (in this course) wewill consider spans of finite (and usuallyrather small) sets of vectors, but a span itself always contains infinitely many vectors (unless the setSconsistsof only the zero vector ).It is often of interest to know whether a particular vector is in the span of a certain set of vectors. The next examplesshow how we do this. Example (c):Isv= 3 2 41 in the span ofS= 1234 , 1 11 1 , 20 31 ?The question is, can we find scalarsc1,c2andc3such thatc1 1234 +c2 1 11 1 +c3 20 31 = 3 2 41 ?

5 (1)We should recognize this as the linear combination form of the system of equations below and to the left. Theaugmented matrix for the system row reduces to the matrix below and to the +c2+ 2c3= 32c1 c2= 23c1+c2 3c3= 44c1 c2+c3= 1 1 0 0 00 1 0 00 0 1 00 0 0 1 This tells us that the system above and to the left has no solution, so there are no scalarsc1,c2andc3for whichequation (1) holds. Thusvis not in the span ofS. Example (d):Isv= 1910 1 in span(S), whereS= 3 12 , 501 , 17 4 ?Here we are trying to find scalarsc1,c2andc3such thatc1 3 12 +c2 501 +c3 17 4 = 1910 1 (2)105We should recognize this as the linear combination form of the system of equations below and to the left. Theaugmented matrix for the system row reduces to the matrix below and to the 5c2+c3= 19 c1+ 7c3= 102c1+c2 4c3= 1 1 0 0 40 1 0 10 0 1 2 This tells us that (2) holds forc1= 4,c2= 1andc3= 2, sovis in span(S).

6 Sometimes, with a little thought, no computations are necessary toanswer such questions, as the next examplesshow. Example (e):Isv= 425 in the span ofS= 302 , 501 ?One can see that any linear combination of the two vectors inSwill have zero as its second component:c1 302 +c2 501 = 3c102c1 + 5c201c2 = 3c1 5c202c1+c2 Since the second component ofvis not zero,vis not in the span of the setS. Example (f):Isv= 47 1 in span 100 , 010 , 001 ?Here we can see that if we multiply the three vectors inSby 4, 7 and 1, respectively, and add them, theresult will bev:4 100 + 7 010 1 001 = 400 + 070 + 00 1 = 47 1 Thereforevis in span 100 , 010 , 001 . Sometimes we will be given an infinite set of vectors, and we ll ask whether a particular finite set of vectorsspansthe infinite set.

7 By this we are asking whether the span of the finite set is the infiniteset. For example, we might askwhether the vectorv= [2,3] spansR2. Because the span of the single vectorvis just a line,vdoes not spanR2. With the knowledge we have at this point, it can sometimes be difficult to tell whether a finite set of vectorsspans a particular infinite set. The next chapter will give us a means for making such a judgement a bit conclude with a few more observations. With a little thought, the following can be seen to be true. (Assumeall vectors are non-zero.) The span of a single vector is all scalar multiples of that vector . InR2orR3the span of a single vector is aline through the origin. The span of a set of two non-parallel vectors inR2is all ofR2.

8 InR3it is a plane through the origin. The span of three vectors inR3that do not lie in the same plane is all Exercises1. Describe the span of each set of vectors inR2orR3by telling what it is geometrically and, if it is a standardset like one of the coordinate axes or planes, specifically what it is. Ifit is a line that is not one of the axes,give two points on the line. If it is a plane that is not one of the coordinate planes, give three points on theplane.(a) The vector 50 inR2.(b) The set of vectors 501 , 003 inR3.(c) The vectors 51 and 03 inR2.(d) The set 000 inR3.(e) The vectors 123 and 246 For each of the following, determine whether the vectorwis in the span of the setS. If it is, write it as alinear combination of the vectors inS.

9 (a)w= 254 ,S= 3 14 , 427 , 3 11 1 (b)w= 5 23128 ,S= 1 4 37 , 26 45 (c)w= 838 1411 ,S= 1 4 37 , 26 45 (d)w= 37 4 ,S= 100 , 110 , 111 107