Orthogonality 3/2/2001 Math 21b, O. KnillON THE RELEVANCE OF ORTHOGONALITY.1) During the pyramid age in Egypt (year -2800 until -2300), the egyptions used ropes divided intolength ratios 3 : 4 : 5 to build triangles. This allowed them to triangulate areas quite precisely:for example to build irrigation (needed because the Nile was shaping the land constantly ) orto build the pyramids: For the great pyramid at Giza with a base length of 230 meters, theaverage error on each side is less then 20 cm, an error of less then 1/1000! A key to achieve thiswas orthogonality2) It was during one of Thales (-624 until -548) journeys to Egypt that he used a geometricaltrick to measure the height of the great pyramid. He measured the size of the shadow of thepyramid. Using a stick, he found the relation between the length of the stick and the length ofits shadow. The same length ratio applies to the pyramid (orthogonal triangles). Thales foundalso that triangles inscribed into a circle and having as the base as the diameter must be orthogonal.3) The Pythagoreans (-572 until -507) were very interested in the discovery that the squares ofa right angle would add up as the Pythagoras theorem a2+ b2= c2tells. They were howeverpuzzled in assigning a length to the diagonal of a square.4) Eratosthenes (-274 until 194) realized that while the sun rays were orthogonal to the groundin the town of Scene, this did no more do so at the town of Alexandria where they would hit theground at 7.2 degrees). Because the distance was about 500 miles and 7.2 is 1/50 of 360 degrees,he measured the circumfence of the earth as 25’000 miles which is pretty close to its actual value.5) Closely related to orthogonality is parallelism. For a long time mathematicians tried toprove Euclid’s parallel axiom using other postulates of Euclid (-325 until -265). These attemptshad to fail because there are geometries in which parallel lines always meet (like on the sphere) orgeometries, where parallel lines never meet (the Poincar´e half plane). Also these geometries canbe studied using linear algebra. The geometry on the sphere with rotations the geometry of thePoincare half plane using complex 2 ×2 matrices.6) The question, whether in reality the angles of a right triangle always add up to 180 degreesbecame a real issue after the discovery of geometries, in which the measurement depends on theposition in space. This Riemannian geometry, founded 150 years ago, is the foundation of generalrelativity a theory describing gravity geometrically: the presence of mass bends space-time.7) In probability theory the notions independence or decorrelated appear. For example,when throwing dice, the number shown by the first dice is independent and therefore decorrelatedfrom the number shown by the second dice. Decorrelation is identical to orthogonality, whenvectors are associated to the random variables.8) In quantum mechanics, states of atoms are described by functions which can be viewed as vec-tors also. The states with energy −EB/n2(where EB= 13.6eV is the Bohr energy) in a hydrogenatom. States in an atom are orthogonal. Two states of two different atoms which don’t interactare orthogonal. One of the challenges in quantum computing, where the computation deals withqubits (=vectors) is that orthogonality is not preserved during the computation. Different statescan interact. This coupling is called decoherence.DEFINITION. Two vectors v and w are called orthogonal if v · w = 0.Examples.1)12is orthogonal to6−3in the plane.2) Both v and w are orthogonal to v × w in three dimensional space.A vector is called a unit vector if its length ||v|| =√v · v is 1.A set of vectors v1, . . . , vnare called orthogonal if they are pairwise orthogonal. They are calledorthonormal if they have additionally length 1. A basis is called an orthonormal basis if theyare are orthogonal and each vector has length 1.Example. For an orthonormal basis, the matrix Aij= vi· vjis the unit matrix.FACTS. Orthogonal vectors are linearly independent. If we have n of them in Rn, then they forma basis.Proof. Multiplying a possible linear relation a1v1+ . . . + anvn= 0 with vkgives akvk· vk=ak||vk||2= 0 and so ak= 0. n linear independent vectors in Rnautomatically span the space.(See the last hour: independence means that the matrix with columns vihas trivial kernel. Thedimension of the image is therefore n by the dimension formula).DEFINITIONS. A vector v is called orthogonal to a linear space V if v is orthogonal to everyvector in V . The orthogonal complement of a linear space V is the set W of all vectors whichare orthogonal to V . It forms a linear space because v·w1= 0, v . . . w2= 0 implies v·(w1+w2) = 0.The orthogonal projection onto a linear space V with orthonormal basis v1, . . . vnis the linearmap A(x) = projV(x) = (v1·x)v1+ . . . + (vn·x)vn. The vector x −projV(x) is in the orthogonalcomplement of V .x and y are always vectors in Rnand V is a linear subspace:PYTHAGORAS: If x and y are orthogonal, then If Pythagoras holds, then x, y are orthogonal.||x + y||2= ||x||2+ ||y||2.PROJECTIONS DO NOT INCREASE LENGTH:||projV(x)|| ≤ ||x|| .(Hint: use Pythagoras) If ||projV(x)|| = ||x||, then x is in V .CAUCHY-SCHWARTZ INEQUALITY:|x · y| ≤ ||x|| ||y|| .(remember the formula for x · y with angle α?) If |x · y| = ||x||||y||, then x and y are parallel.TRIANGLE INEQUALITY:||x + y|| ≤ ||x|| + ||y||because (x + y) · (x + y) = ||x||2+ ||y||2+ 2x · y ≤ ||x||2+ ||y||2+ 2||x||||y|| = (||x|| + ||y||)2,DEFINITION. The angle between two vectors x, y isα = arccosx · y||x||||y||.Example. The angle between two orthogonal vectors is 90 degrees or 270 degrees. If x and yrepresent data showing the deviation from the mean, thenx·y||x||||y||is called the correlation of thedata. The extremest positive correlation is when x = y.A QUESTION.Express the fact that x is in the kernel of a matrix A using orthogonality of row/column vectorsin A.ANSWER: Ax = 0 means that wk· x = 0 for every row vector wkof Rn.REMARK. We will call later the matrix AT, obtained by switching rows and columns of A thetranspose of A. You see already that the image of ATis orthogonal to the kernel of A.A QUESTION: Find a basis for the orthogonal complement of the linear space V spanned by1234. and4567.ANSWER: The orthogonality ofxyzuto the two vectors means solving the linear system ofequations x +
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