By Jorge Rebaza

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Extra info for A First Course in Applied Mathematics
Example text
That is, N(A) is a line spanned by the vector [—1 2 — 1] T . 47). 59 "2 0 1 3 0" 0 3 1 2 0 In this case the columns 1, 2, and 5 are linearly indeLet A = 2 3 2 5 1 0 0 0 0 0 pendent and they form a basis of col( A). That is, any vector in col(^4) (in particular, the third and fourth columns of ^4) can be expressed as a unique combination of columns 1, 2, and 5. This means that geometrically, col(A) is a 3dimensional hyperplane (in M4) spanned by those three columns, and therefore N(A) is a twodimensional subspaceof M5.
Proof. , um} be an orthonormal basis of U. 40), let U = (vTUi)ui \ 1 {^Furr^Um Then, the vector w = v — u is orthogonal to each Ui, i — 1 , . . , m, and therefore, it is in U1. Hence, v = u + w, where u G U and w G U^. Also, it is clear that the only intersection ofU and U1 is the zero vector, for ifueU and u G U^, then \\u\\2 =uTu = 0. This proves that in fact, V = U 0 U^. 52 If V = R 3 , and U is the XFplane, then U1 is the Zaxis. Then, it is clear that V = U 0 UL. In fact, every vector [x y z]T can be uniquely written as [x y z)T = [x y 0] T + [0 0 z]T.
63 Let S be the subspace of R 3 spanned by u\ = [2 1  1] T and u2 = [0  1 1] T . Then the matrix 4 0 0 2 1 1 P = 2 3 3 is a projection matrix onto S. 62: P not a projection. The projection matrices that probably have more applications are those that are also orthogonal. 54 An orthogonal projection matrix is a projection matrix P for which PT — P. 64 The matrix P = 1/2 1/2 1/2 1/2 is an orthogonal projection. It clearly satisfies P2 = P and PT = P. 62). 55 It is important to note that an orthogonal projection matrix is not necessarily an orthogonal matrix.