Math 311-504Topics in Applied MathematicsLecture 7:Linear independence (continued).Matrix algebra.Linear independenceDefinition. Vectors v1, v2, . . . , vk∈ Rnare calledlinearly dependent if they satisfy a relationt1v1+ t2v2+ · · · + tkvk= 0,where the coefficients t1, . . . , tk∈ R are not allequal to zero. Otherwise the vectors v1, v2, . . . , vkare called linear ly independent. That is, ift1v1+t2v2+ · · · +tkvk= 0 =⇒ t1= · · · = tk= 0 .Theorem The vectors v1, . . . , vkare linearlydependent if and only i f one of them is a linearcombination of the others.Definition. A subset S ⊂ Rnis called a hyperplane(or an affine subspace) if it has a parametricrepresentation t1v1+ t2v2+ · · · + tkvk+ v0,where viare fixed n-dimensional vectors and tiarearbitrary scalars.The number k of parameters may depend on arepresentation. The hyperplane S is called ak-plane if k is as small as possible.Theorem A hyperpl anet1v1+ t2v2+ · · · + tkvk+ v0is a k-plane if and only if vectors v1, v2, . . . , vkarelinearly independent.Examples• Vectors e1= (1, 0, 0), e2= (0, 1, 0), ande3= (0, 0, 1) in R3.t1e1+ t2e2+ t3e3= 0 =⇒ (t1, t2, t3) = 0=⇒ t1= t2= t3= 0Thus e1, e2, e3are linearly independent.• Vectors v1= (4, 3, 0, 1), v2= (1, −1, 2, 0), andv3= (−2, 2, −4, 0) in R4.It is easy to observe that v3= −2v2.=⇒ 0v1+ 2v2+ 1v3= 0Thus v1, v2, v3are linearly dependent. At the same time, thevector v1is not a linear combination of v2and v3.• Vectors u1= (1, 2, 0), u2= (3, 1, 1), andu3= (4, −7, 3) in R3.We need to check if the vector equation t1u1+ t2u2+ t3u3= 0has solutions other than t1= t2= t3= 0.This vector equation is equivalent to a systemr1+ 3r2+ 4r3= 0,2r1+ r2− 7r3= 0,r2+ 3r3= 0.1 3 402 1 −700 1 30Row reduction yields:1 3 402 1 −7 00 1 3 0→1 3 400 −5 −15 00 1 3 0→1 3 4 00 1 3 00 0 0 0The variable t3is free =⇒ there are infinitely many solutions=⇒ the vectors u1, u2, u3are linearly dependent.MatricesDefinition. An m-by-n matrix is a rectangulararray of numbers that has m row s and n columns:a11a12. . . a1na21a22. . . a2n............am1am2. . . amnNotation: A = (aij)1≤i≤n, 1≤j≤mor simply A = ( aij)if the dimensions are known.An n-dimens ional vector can be represented as a1 × n matri x (row vector) or as an n × 1 matrix(column vector):(x1, x2, . . . , xn)x1x2...xnAn m × n m atrix A = (aij) can be regarded as acolumn of n-dimensional row vectors or as a row o fm-dimensional col umn vectors:A =v1v2...vm, vi= (ai1, ai2, . . . , ain)A = (w1, w2, . . . , wn), wj=a1ja2j...amjVector algebraLet a = (a1, a2, . . . , an) and b = (b1, b2, . . . , bn)be n-dimensional vectors, and r ∈ R be a scalar.Vector sum: a + b = (a1+ b1, a2+ b2, . . . , an+ bn)Scalar multiple: ra = (ra1, ra2, . . . , ran)Zero vector: 0 = (0, 0, . . . , 0)Negative of a vector: −b = (−b1, −b2, . . . , −bn)Vector di fference:a − b = a + (−b) = (a1− b1, a2− b2, . . . , an− bn)Matrix algebraDefinition. Let A = (aij) and B = (bij) be m×nmatrices. The sum A + B is defined to be the m×nmatrix C = (cij) such that cij= aij+ bijfor allindices i, j.That is, two matrices with the same dimensions canbe added by adding their corresponding entries.a11a12a21a22a31a32+b11b12b21b22b31b32=a11+ b11a12+ b12a21+ b21a22+ b22a31+ b31a32+ b32Definition. Given an m×n matrix A = ( aij) and anumber r , the scalar multiple rA is defined to bethe m×n matrix D = (dij) such that dij= raijfor allindices i, j.That is, to multiply a matrix by a scalar r ,one multiplies each entry of the matrix by r.ra11a12a13a21a22a23a31a32a33=ra11ra12ra13ra21ra22ra23ra31ra32ra33The m×n zero matrix (all entries are zeros) isdenoted Omnor simply O.Negative of a matrix: −A is defined as (−1)A.Matrix difference: A − B is defined as A + (−B).As far as the linear oper ati ons (addition and scalarmultiplication) are concerned, the m×n matricescan be regarded as mn-dimensio nal vectors.ExamplesA =3 2 −11 1 1, B =2 0 10 1 1,C =2 00 1, D =1 10 1.A + B =5 2 01 2 2, A − B =1 2 −21 0 0,2C =4 00 2, 3D =3 30 3,2C + 3D =7 30 5, A + D is not defined.Properties o f linear operations(A + B) + C = A + (B + C )A + B = B + AA + O = O + A = AA + ( −A) = (−A) + A = Or(sA) = (rs)Ar(A + B) = rA + rB(r + s)A = rA + sA1A = A0A = ODot productDefinition. The dot product of n-dimensionalvectors x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn)is a scalarx · y = x1y1+ x2y2+ · · · + xnyn=nXk=1xkyk.Matrix multiplicationThe product of matrices A and B is defined if thenumber of columns in A matches the number ofrows in B.Definition. Let A = (aik) be an m×n matrix andB = (bkj) be an n×p matri x. The product AB isdefined to be the m×p matrix C = (cij) such thatcij=Pnk=1aikbkjfor all indices i, j.That is, matrices are multiplied r ow by column:∗ ∗ ∗* * *∗ ∗* ∗∗ ∗ * ∗∗ ∗ * ∗=∗ ∗ ∗ ∗∗ ∗ * ∗A =a11a12. . . a1na21a22. . . a2n............am1am2. . . amn=v1v2...vmB =b11b12. . . b1pb21b22. . . b2p............bn1bn2. . . bnp= (w1, w2, . . . , wp)=⇒ AB =v1·w1v1·w2. . . v1·wpv2·w1v2·w2. . . v2·wp............vm·w1vm·w2. . . vm·wpExamples.(x1, x2, . . . , xn)y1y2...yn= (Pnk=1xkyk),y1y2...yn(x1, x2, . . . , xn) =y1x1y1x2. . . y1xny2x1y2x2. . . y2xn............ynx1ynx2. . . ynxn.Example.1 1 −10 2 10 3 1 1−2 5 6 01 7 4 1=−3 1 3 0−3 17 16 1Any system of linear equations can be rewritten as amatrix equation.a11x1+ a12x2+ · · · + a1nxn= b1a21x1+ a22x2+ · · · + a2nxn= b2· · · · · · · · ·am1x1+ am2x2+ · · · + amnxn= bm⇐⇒a11a12. . . a1na21a22. . . a2n............am1am2. . . amnx1x2...xn=b1b2...bmProperties o f m atrix multiplication:(AB)C = A(BC ) (associative law)(A + B)C = AC + BC (distributive law #1)C (A + B) = CA + CB (distributive law #2)(rA)B = A(rB) = r (AB)(Any of the above identities holds provided thatmatrix sums and