Linear Algebra (2nd Edition)
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This introduction to linear algebra features intuitive introductions and examples to motivate important ideas and to illustrate the use of results of theorems.
Linear Equations; Vector Spaces; Linear Transformations; Polynomials; Determinants; Elementary canonical Forms; Rational and Jordan Forms; Inner Product Spaces; Operators on Inner Product Spaces; Bilinear Forms
For all readers interested in linear algebra.
Control Theory for Partial Differential Equations: Volume 2, Abstract Hyperbolic-like Systems over a Finite Time Horizon: Continuous and Approximation Theories (Encyclopedia of Mathematics and Its Applications Series, Book 75)
enormously the verification that T is 1: 1. Let us call a linear transformation T non-singular if Ty = 0 implies 79 80 Linear Transformations Chap. 3 y = 0, i.e., if the null space of T is (0). Evidently, T is 1: 1 if and only if T is non-singular. The extension of this remark is that non-singular linear transformations are those which preserve linear independence. Theorem T is non-singular V onto a linearly 8. Let T be a linear transformation from V into W. Then if and only if T carries
Let dl,..., d, be a finite number of polynomials over F. Then the sum M of the subspaces dzF[x] is a subspace and is also an ideal. For suppose p belongs to M. Then there exist polynomials fl, . . . , fn in polynomial F [r] such that p = dlfl + * . . + dnfn. If g is an arbitrary over F, then pg = dl(flg) + . . 1 + dn(fng) so that pg also belongs to M. Thus M is an ideal, and we say that M is the ideal generated by the polynomials, dl, . . . , d,. EXAMPLE 7. Let F be a subfield sider the ideal M
a formula for A-l. if and only 5. Let A be a 2 X 2 matrix over a field F, and suppose that A2 = 0. Show for each scalar c that det (cl - A) = c2. 6. Let K be a subfield of the complex numbers and n a positive integer. Let 311 f * *, j, and ICI, . . . , k, be positive integers not exceeding n. For an n X n matrix A over K define D(A) = A($, k3A(jz,k2) . . . ALL, W. Prove that D is n-linear if and only if the integers jr, . . . , j,, are distinct. 7. Let K be a commutative ring with identity.
proof of that fact is an interesting application of determinants. 5. Let K be a commutative ring with identity. with n generators, then the rank of V is n. Theorem K-module If V is a free Proof. We are to prove that V cannot be spanned by less than n of its elements. Since V is isomorphic to Kn, we must show that, if m < n, the module KS is not spanned by n-tuples acl, . . . , (Y,. Let A be the matrix with rows al, . . . , CY,. Suppose that each of the standard basis vectors cl, . . . , en is
j, < . . 1 < j,. This special type of r-tuple is called an r-shuffle of (1, . . . , n}. There are n 0r n! = r!(n - r)! such shuffles. Suppose we fix an r-shuffle J. Let LJ be the sum of all the terms in (5-37) corresponding to permutations of the shuffle J. If (Tis a permutation of (1, . . . , r}, then L(Pjm * . . ) Pj.3 = (w u> L@ji, . . - 7 &I* Thus (5-38) LJ = L(Pju . . . 9Pj,lDJ where (5-39) DJ = 2 (en u>fj.1 0 * * * 0 fj.. L7 = rv(fjl 0 * ’ * 0 fj,) * We see from (5-39) that