By Jorge Rebaza
ISBN10: 1118229622
ISBN13: 9781118229620

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Additional resources for A First Course in Applied Mathematics
Example text
19 In the complex case, we say a matrix Umxn is unitary if U*U = I, 18 BASICS OF LINEAR ALGEBRA where as before U* denotes the adjoint ofU. 26) and the statements (a) (d) above, immediately set orthogonal matrices apart from most other matrices, and it would be enough to consider them very important in theory and applications. But there is more. A very important and useful fact about orthogonal matrices is that they preserve the norm of a vector or a matrix. More precisely, we have the following theorem.
38 Observe that each column of AmXn why the col(A) is a subspace ofM™. 37 and (L43), we conclude that for any x G Rn, Ax is a combination of the columns of A. 40 The dimension of the column space of a matrix Amxn (the number of linearly independent columns of A) is called the rank of A. • 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.
Vn} is said to be a basis of a vector space V if it spans V and is linearly independent. The most important application of this definition is that if x is an arbitrary element of V, then it can be written in a unique form as a linear combination of the elements of the basis. 36) for some unique scalars c i , . . , c n . 36) uniquely determine the vector a: on a given basis. For a different basis, there will be different coefficients, but again this combination is unique, on that basis. 30 we can say that a basis B is a genuine (although not unique) representative of a vector space V.
A First Course in Applied Mathematics by Jorge Rebaza
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