The geometry of l inear equationsAdding and scaling vectorsExample 1. We have already encountered ma trices such as1 4 2 32 −1 2 23 2 −2 0.Each column is what we call a (column) vector.In this example, each column vector has 3 entries and so lies in R3.Example 2. A fundamental property of vectors is that vectors o f t h e same kind can beadded and sc aled.123+4−12= , 7 ·x1x2x3= .Example 3. (Geometric description of R2) A vectorx1x2represents the point(x1, x2) in the plan e .Givenx =13and y =21, graph x, y, x + y, 3x.Adding and scaling vectors, the most general th ing we can do is:Definition 4. Given vectors v1, v2,, vmin Rnand scalars c1, c2,, cm, the vect orc1v1+ c2v2++ cmvmis a linear combination of v1, v2,, vm.The scalarsc1,, cmare the coefficients or weights.Example 5. Linear combinations of v1, v2, v3include:• 3v1− v2+ 7v3,• v2+ v3,•13v2,• 0.Example 6. Express15as a linear combination of21and−11.Solution. We have to find c1and c2such thatArmin [email protected] row and column pictureExample 7. We can thi n k of the linear system2x − y = 1x + y = 5in two different geometric ways.Row picture.Each equati on defines a l ine in R2.Which points lie on the intersection of thes e lines?Column picture.The system can be writt e n as x21+ y−11=15.Which linear combinations of21and−11produce15?This example has the unique solutionx = 2, y = 3.• (2, 3) is the (only) intersection of the two lines 2x − y = 1 and x + y = 5.• 221+ 3−11is the (only) linear combination producing15.Example 8. Consider the vectorsa1=103, a2=4214, a3=3610, b =−18−5.Determine if b is a linear combination of a1, a2, a3.Solution. Vector b is a linear combination of a1, a2, a3if we can find weights x1, x2,x3such that:This vector equation correspo nds to the linear syste m:Armin [email protected] augm e nted matrix:Row reduction to echelon form:Hence:Example 9. In the previous example, express b as a linear combination of a1, a2, a3.Solution.SummaryA vector equationx1a1+ x2a2++ xmam= bhas the same solution set as the linear system with augmented matrix| | | |a1a2amb| | | |.In particular, b can be generated by a linear combination of a1, a2,, amif and onlyif there is a solution to this linear system.Armin [email protected] span of a set of vectorsDefinition 10. The span of vectors v1, v2,, vmis th e set of all their linear combina-tions. We denote it by span{v1, v2,, vm}.In other words,span{v1, v2,, vm} is the set of all vectors of the formc1v1+ c2v2++ cmvm,where c1, c2,, cmare scalars.Example 11.(a)Describe spann21ogeometrica lly.(b) Describe spann21,41ogeometrica lly.(c) Describe spann21,42ogeometrica lly.A single (nonzero) vector always spans a line, two vectorsv1, v2usually span a planebut it could a lso be just a line ( if v2= αv1).We will come back to this when we discuss dimension and linear independence.Example 12. Is span(2−11,4−21)a line or a plane?Solution.Example 13.ConsiderA =1 23 10 5, b =8317.Is b in the p lane spanned by the columns of A?Solution.Armin [email protected] an d summary• The span of vectors a1, a2,, amis the set of all their linear combination s.• Some vector b is in span{a1, a2,, am} if and only if there is a solu t ion to thelinear system with augmented matrix| | | |a1a2amb| | | |.◦ Each solution c orresponds to the weights in a linear combination of the a1, a2,,amwhich gives b.◦ This gi ves a second geometr ic way to think of linear systems!Armin
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