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MIT 12 215 - Basic vectors and matrices

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112.215 Modern NavigationThomas Herring ([email protected]),http://geoweb.mit.edu/~tah/12.21510/14/2009 12.215 Modern Naviation L09 2Review of last class• Sextant measurements using the sun:– We tracked the sun to find its highest elevation andthe time this occurs.• Our one cheat was using computer NTP to get time(We will use GPS to check the results)210/14/2009 12.215 Modern Naviation L09 3Todayʼs Class• Review of linear Algebra. Class will be based on thebook “Linear Algebra, Geodesy, and GPS”, G. Strangand K. Borre, Wellesley-Cambridge Press, Wellesley,MA, pp. 624, 1997• Topics to be covered will be those later in the course• General areas are:– Vectors and matrices– Solving linear equations– Vector Spaces– Eigenvectors and values– Rotation matrices10/14/2009 12.215 Modern Naviation L09 4Basic vectors and matrices• Important basic concepts• Vectors: A column representing a set of n-quantities– In two and three dimensions these can be visualized asarrows between points with different coordinates with thevector itself usually having on end at the origin– The same concept can be applied to any n-dimensional vector– Vectors can be added and subtracted (head-to-tail) by addingand subtracting the individual components of the vectors.– Linear combinations of vectors can be formed by scaling andaddition. The result is another vector e.g., cv+dw– (Often a bold symbol will be used to denote a vector and sometimes a line is drawn over the top).310/14/2009 12.215 Modern Naviation L09 5Lengths and dot products• The dot product or inner product of two vectors isdefined as:• The order of the dot product makes no difference.• The length or norm of a vector is the square-root ofthe dot product• A unit vector is one with unit length• If the dot product of two vectors is zero they are saidto be orthogonal (orthonormal if they are unit vectors)• The components of a 2-D unit vector are cos and sinof the angle the vector makes to the x-axis. € v ⋅ w = v1w1+ v2w2+ L + vnwn10/14/2009 12.215 Modern Naviation L09 6Angles between vectors• The cosine formula:• Schwarz inequality: The dot product of any twovectors is less or equal the product of the lengths ofthe two vectors• The representation of a plane is by the vector that isnormal to it. For a plane through the origin, all vectorsin that plane must have zero dot product with thenormal. This then provides an equation for the plane.€ cosθ=v ⋅ w| v || w |410/14/2009 12.215 Modern Naviation L09 7Planes• If a plane does not contain the origin, then thecoordinates of a point on the normal containing theplane specifies the plane. The dot product of thenormal and points in the plane then is a constant.• If the normal to the plane is a expressed as a unitvector, the constant is the closest distance of theplane to the origin.• In N-dimensional space, the concept of a plane is thesame: it is defined by n.v=d• Any two non-collinear vectors define a plane10/14/2009 12.215 Modern Naviation L09 8Matrices and linear equations• Any set of linear equations (i.e., equations which donot contain powers or products of the unknowns) canbe written in matrix form with the coefficients of thelinear equations being the elements of the matrix.• The rows and columns of matrices are themselvesvectors.• A matrix represents a linear combination of theelements of a vector to form another vector of possiblydifferent length:€ Ax = b where x and b are vectors of length n and mA is a m - rows and n column matrix510/14/2009 12.215 Modern Naviation L09 9Solving linear equations• If the x and b vectors are the same length, then givenA and b it is often possible to find x (sometimes this isnot possible and sometimes x may not be unique).• There a many methods for solving this type of systembut the earliest ones are by elimination i.e., linearcombinations are formed of the rows of the matrix Athat eliminate one of the elements of x. The processis repeated until only one element of x remains (whichis easily solved). Back substitution allows all thecomponents of x to be computed.• This process is sometimes viewed as multiplying byeliminator matrices.10/14/2009 12.215 Modern Naviation L09 10Rules of matrix multiplication• The product of two matrices A (n-rows and m-columns)by B (r-rows and c-columns) is only possible if m=r• The resultant matrix has n-rows and c-columns.• In general, AB does not equal BA even the matricesare square• A matrix multiplication is the dot products of rows ofthe first matrix with the columns of the second matrix• Matrix multiplication is associative (AB)C=A(BC) butnot commutative• A matrix is invertible if A-1 such that A-1A=I where I isa unit matrix, exists.610/14/2009 12.215 Modern Naviation L09 11Factorization• In factorization a matrix A is written as A=LU where isL is a lower triangular matrix and U is an uppertriangular matrix.• The individual matrices L and U are not unique (L canbe multiplied by a scalar and U divided by the samescalar with out changing the product. Convention hasthe diagonal of L being 1ʼs.• Why factorize? Since forms are lower triangular,substitute down (L) and up (U) the matrix€ Solve Lc = b then solve Ux = c to solve Ax = b10/14/2009 12.215 Modern Naviation L09 12Characteristics of LU• When the rows of A start with zero so do thecorresponding rows of L; when the columns of A startwith 0 so do the columns of U.• Many estimation problems are “band-limited” i.e., onlya small number of the elements around the diagonalare non-zero; the L and U matrices will also be band-limited but the inverse of such a matrix is normally full.(Factorization saves time and space).• http://web.mit.edu/18.06/www/Course-Info/Mfiles/slu.m is a linkto an SLU matlab code (also code at same site thatpivots the matrix which is a more stable approach).710/14/2009 12.215 Modern Naviation L09 13Transpose• The transpose of a matrix is the matrix with rows andcolumns switched. Usually denoted as AT orsometimes Aʼ• Some rules: (AB)T=BTAT• A symmetric matrix is one for which A=AT• The products ATA and AAT generate symmetricmatrices. We will see these forms many times in whenwe cover estimation and statistics.10/14/2009 12.215 Modern Naviation L09 14Matrix rank• The rank of a matrix is the number of non-zero pivotsin the matrix. Pivots are the number you divide bywhen doing Gauss elimination or when solving the ULsystem.• The rank is the


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