ORTHOGONALITY Math 21b, O. KnillORTHOGONALITY. ~v and ~w are called orthogonal if ~v · ~w = 0.Examples. 1)12and6−3are orthogonal in R2. 2) ~v and w are both orthogonal to ~v × ~w in R3.~v is called a unit vector if ||~v|| =√~v ·~v = 1. B = {~v1, . . . , ~vn} are called orthogonal if they are pairwiseorthogonal. They are called orthonormal if they are also unit vectors. A basis is called an orthonormalbasis if it is orthonormal. For an orthonormal basis, the matrix Aij= ~vi·~vjis the unit matrix.FACT. Orthogonal vectors are linearly independent and n orthogonal vectors in Rnform a basis.Proof. The dot product of a linear relation a1~v1+ . . . + an~vn= 0 with ~vkgives ak~vk·~vk= ak||~vk||2= 0 sothat ak= 0. If we have n linear independent vectors in Rnthen they automatically span the space.ORTHOGONAL COMPLEMENT. A vector ~w ∈ Rnis called orthogonal to a linear space V if ~w is orthogonalto every vector in ~v ∈ 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.ORTHOGONAL PROJECTION. The orthogonal projection onto a linear space V with orthnormal basis~v1, . . . , ~vnis the linear mapT (~x) = projV(x) = (~v1·~x)~v1+ . . . + (~vn·~x)~vnThe vector ~x − projV(~x) is in theorthogonal complement of V . (Note that ~viin the projection formula are unit vectors, they have also to beorthogonal.)SPECIAL CASE. For an orthonormal basis ~vi, one can write~x = (~v1·~x)~v1+ ... + (~vn·~x)~vn.PYTHAGORAS: If ~x and ~y are orthogonal, then ||~x + ~y||2= ||~x||2+ ||~y||2. Proof. Expand (~x + ~y) · (~x + ~y).PROJECTIONS DO NOT INCREASE LENGTH: ||projV(~x)|| ≤ ||~x||. Proof. Use Pythagoras: on ~x =projV(~x) + (~x − projV(~x))). If ||projV(~x)|| = ||~x||, then ~x is in V .CAUCHY-SCHWARTZ INEQUALITY: |~x · ~y| ≤ ||~x|| ||~y|| . Proof: ~x ·~y = ||~x||||~y||cos(α).If |~x ·~y| = ||~x||||~y||, then ~x and ~y are parallel.TRIANGLE INEQUALITY: ||~x + ~y|| ≤ ||~x|| + ||~y||. Proof: (~x + ~y) · (~x + ~y) = ||~x||2+ ||~y||2+ 2~x · ~y ≤||~x||2+ ||~y||2+ 2||~x||||~y|| = (||~x|| + ||~y||)2.ANGLE. The angle between two vectors ~x, ~y isα = arccos~x ·~y||~x||||~y||.CORRELATION. cos(α) =~x·~y||~x||||~y||is called the correlation between ~xand ~y. It is a number in [−1, 1].EXAMPLE. The angle between two orthogonal vectors is 90 degrees or 270 degrees. If ~x and ~y represent datashowing the deviation from the mean, then~x·~y||~x||||~y||is called the statistical correlation of the data.QUESTION. Express the fact that ~x is in the kernel of a matrix A using orthogonality.ANSWER: A~x = 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 the transpose ofA. You see already that the image of ATis orthogonal to the kernel of A.QUESTION. Find a basis for the orthogonal complement of the linear space V spanned by1234,4567.ANSWER: The orthogonality ofxyzuto the two vectors means solving the linear system of equations x +2y + 3z + 4w = 0, 4x + 5y + 6z + 7w = 0. An other way to solve it: the kernel of A =1 2 3 44 5 6 7is theorthogonal complement of V . This reduces the problem to an older problem.ON THE RELEVANCE OF ORTHOGONALITY.1) During the pyramid age in Egypt (from -2800 til -2300 BC),the Egyptians used ropes divided into length ratios 3 : 4 : 5 tobuild triangles. This allowed them to triangulate areas quite pre-cisely: for example to build irrigation needed because the Nile wasreshaping the land constantly or to build the pyramids: for thegreat pyramid at Giza with a base length of 230 meters, theaverage error on each side is less then 20cm, an error of less then1/1000. A key to achieve this was orthogonality.2) During one of Thales (-624 til -548 BC) journeys to Egypt,he used a geometrical trick to measure the height of the greatpyramid. He measured the size of the shadow of the pyramid.Using a stick, he found the relation between the length of thestick and the length of its shadow. The same length ratio appliesto the pyramid (orthogonal triangles). Thales found also thattriangles inscribed into a circle and having as the base as thediameter must have a right angle.3) The Pythagoreans (-572 until -507) were interested in the dis-covery that the squares of a lengths of a triangle with two or-thogonal sides would add up as a2+ b2= c2. They were puzzledin assigning a length to the diagonal of the unit square, whichis√2. This number is irrational because√2 = p/q would implythat q2= 2p2. While the prime factorization of q2contains aneven power of 2, the prime factorization of 2p2contains an oddpower of 2.4) Eratosthenes (-274 until 194) realized that while the sun rayswere orthogonal to the ground in the town of Scene, this did nomore do so at the town of Alexandria, where they would hit theground at 7.2 degrees). Because the distance was about 500 milesand 7.2 is 1/50 of 360 degrees, he measured the circumference ofthe earth as 25’000 miles - pretty close to the actual value 24’874miles.5) Closely related to orthogonality is parallelism. For a long time mathematicianstried to prove Euclid’s parallel axiom using other postulates of Euclid (-325 until -265).These attempts had to fail because there are geometries in which parallel lines alwaysmeet (like on the sphere) or geometries, where parallel lines never meet (the Poincar´ehalf plane). Also these geometries can be studied using linear algebra. The geometry onthe sphere with rotations, the geometry on the half plane uses M¨obius transformations,2 × 2 matrices with determinant one.6) The question whether the angles of a right triangle are in reality always add up to180 degrees became an issue when geometries where discovered, in which the measure-ment depends on the position in space. Riemannian geometry, founded 150 years ago,is the foundation of general relativity a theory which describes gravity geometrically:the presence of mass bends space-time, where the dot product can depend on space.Orthogonality becomes relative too.7) In probability theory the notion of independence or decorrelation is used. Forexample, when throwing a dice, the number shown by the first dice is independent anddecorrelated from the number shown by the second dice. Decorrelation is identical toorthogonality, when vectors are
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