Orthogonality Review 3/8/2002 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 orthonormal basis~v1, . . . ,~vnis the linear map T(~x) = projV(x) = (~v1· ~x)~v1+ . . . + (~vn· ~x)~vn. The vector ~x − projV(~x) is in theorthogonal complement of V . (Note that ~viin the projection formula are unit vectors!)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 be-tween ~x and ~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 (year -2800 BC until -2300BC), the Egyptians used ropes divided into length ratios 3 : 4 : 5to build triangles. This allowed them to triangulate areas quiteprecisely: for example to build irrigation (the Nile was reshapingthe land constantly) or to build the pyramids: For the greatpyramid at Giza with a base length of 230 meters, the averageerror on each side is less then 20cm, an error of less then 1/1000.A key to achieve this was orthogonality.2) During one of Thales (-624 BC until -548 BC) journeys toEgypt, he used a geometrical trick to measure the height ofthe great pyramid. He measured the size of the shadow of thepyramid. Using a stick, he found the relation between the lengthof the stick and the length of its shadow. The same length ratioapplies to the pyramid (orthogonal triangles). Thales found alsothat triangles 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 thediscovery that the squares of a right angle would add up asa2+ b2= c2. They were puzzled in assigning a length to thediagonal of a square.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. This is pretty close to its actual value24’874 miles.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 linesalways meet (like on the sphere) or geometries, where parallel lines never meet (thePoincar´e half plane). Also these geometries can be studied using linear algebra. Thegeometry on the sphere with rotations, the geometry of the Poincare half plane uses2 × 2 matrices.6) The question, whether in reality the angles of a right triangle always add up to180 degrees became a real issue when geometries where discovered, in which themeasurement depends on the position in space. Riemannian geometry, founded 150years ago, is the foundation of general relativity, a theory which describes gravitygeometrically: the presence of mass bends space-time.7) In probability theory the notions independence or decorrelated appear. Forexample, when throwing dice, the number shown by the first dice is independent andtherefore decorrelated from the number shown by the second dice. Decorrelation isidentical to orthogonality, when vectors are associated to the random variables.8) In quantum mechanics, states of atoms are described by functions which can beviewed as vectors also. The states with energy −EB/n2(where EB= 13.6eV is theBohr energy) in a hydrogen atom. States in an atom are orthogonal. Two statesof two different atoms which don’t interact are orthogonal. One of the challengesin quantum computing, where the computation deals with qubits (=vectors) is thatorthogonality is not preserved
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