By Jorge Rebaza
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Additional resources for A First Course in Applied Mathematics
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 2-dimensional; 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