By Jorge Rebaza
ISBN10: 1118229622
ISBN13: 9781118229620

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Extra resources for A First Course in Applied Mathematics
Example text
However, we will use orthogonal matrices to construct orthogonal projection matrices. Explicit formulas for orthogonal projections. We want to see how to obtain an explicit expression for an orthogonal projection onto a subspace £, starting with the particular case when S — col(A). Later we will see that for a general m x n matrix A, with m > n, the least squares solution to the system Ax = b is given by the solution of the socalled normal equations ATAx = ATb. This is nothing else but a consequence of projecting b orthogonally onto col(A), as it will be explained in detail in Chapter 4.
MATLAB command: rank(A). 54 LetA = 1 5 4 2 3 7 Ax = be arbitrary. 39. We also observe that in this case, since the two columns of A are linearly independent, the col(A) is 2dimensional; that is, geometrically it is a plane in M3 spanned by the two columns of the matrix A, and thus, rank( A) = 2. 55 ["21 0 0 1  1 . By definition, the co\(A) is the set of vectors of the [2 0  1 form Ax, for any x. 55 that although it is true that the col(A) is spanned by (all) the columns of A, and A has three columns, they are not linearly independent.
G. 2. 2 Householder reflection of x. 2 is a reflection of x with respect to u x . 30) Orthonormal extensions. One important application of Householder matrices is extending a given vector x G R n to a set of n orthonormal vectors. The idea is to get an orthogonal matrix H, with its first column being the vector x, normalized, if necessary. , when proving several theorems in Chapter 3. Let x £ R n be a vector with llxll2 = xTx — 1. Define the vector u = x — e\. T T First observe that u x = x x — efx — 1 — ejx and that uTu = xTx — xTe\ — ejx + ejei = 2(1 — ejx), so that uTu = 2uTx.
A First Course in Applied Mathematics by Jorge Rebaza
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