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Additional info for A First Course in Applied Mathematics
39), we get r '^CLiUi *=1 m + ^^bjVj = 0. j= l Similarly, since By is linearly independent, we must have a; = 0 and 6j = 0, for all z = 1 , . . , r and j = 1 , . . , m. Thus, B is linearly independent and therefore a basis of V + W. Finally, using the definition of dimension, we get dim(F + l^) = r + ra + n = (r + ra) + (r + n ) - r = dimF + dimW - dim(V fl W). 5 Gram-Schmidt: W2 is orthogonal to q\. 7 ORTHOGONALIZATION AND DIRECT SUMS Given a set of linearly independent vectors, it is always possible to transform it into an orthonormal set.
47) Proof. The vector spaces col(A) and N(AT) are subspaces of R™. Similarly, the vector spaces col(AT) and N(A) are subspaces of Rn. 46. 47 is the following corollary. A T ) = r, dimN(A) = n-r, dimiV(^4T) = m-r. 1 implies the following important result. 48 (Dimension Formula). Consider anm x n matrix A. Then, dimcol(A) + dimAT(^l) = n. 58 A = ri 4 7i 2 5 8 . Observe that we can write the third column a^ as a combi|_3 6 9J nation of the first two oi, 0,2 as as = 2a,2 — a\. And since these first two columns are linearly independent, this implies that co\(A) is 2-dimensional.
If A = (a^), then we can also say A is symmetric if a^ = a^, for i,j = l , . . , n . 12 For an arbitrary matrix A, the matrices A + AT and A AT are symmetric. is complex, we say A is Hermitian if A = A*. 15 IfAnXn following matrix is Hermitian: 1 4-2z 4 + 2% - 9 In the set of real numbers, we can define a number a to be positive if a x2 > 0, for all numbers x ^ 0. The generalization of this definition for matrices leads to the so-called positive definite matrices. 23) for all vectors x ^ 0. If instead we have xT A x > 0, then the matrix is said to be positive semidefinite.
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