Linear Algebra, Second Edition by Kenneth M Hoffman, Ray Kunze
By Kenneth M Hoffman, Ray Kunze
This creation to linear algebra gains intuitive introductions and examples to encourage very important rules and to demonstrate using result of theorems.
Linear Equations; Vector areas; Linear variations; Polynomials; Determinants; easy canonical types; Rational and Jordan types; internal Product areas; Operators on internal Product areas; Bilinear Forms
For all readers drawn to linear algebra.
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Additional info for Linear Algebra, Second Edition
Since under an elementary operation of type II neither F (A) nor |A| is changed, and under elementary operations of type I, both expressions change sign, it follows that |A| = (−1)t |A|, F (A) = (−1)t F (A). 25), that is, to establish the equality F (A) = k|A|, which, in turn, clearly follows from the relationships |A| = a 11 a 22 · · · a nn , F (A) = F (E) · a 11 a 22 · · · a nn . 20) from the previous section. Moreover, it is a consequence of the second equality, since the determinant |A|, as we have shown, is also a function of type F (A), possessing properties 1 and 2.
20), which is a system of linear equations. The numbers a0 , . . , an are the unknowns. The number of unknowns 16 1 Linear Equations is n + 1 (the numeration begins here not with the usual a1 , but with a0 ). The numbers 1 and cik are the coefficients of the unknowns, and k1 , . . , kr are the constant terms. 12 and its corollary. 20) has only the null solution. This associated system can be written in the form ⎧ f (c1 ) = 0, ⎪ ⎪ ⎪ ⎨ f (c2 ) = 0, ⎪ ··· ⎪ ⎪ ⎩ f (cr ) = 0. 21) A number c for which f (c) = 0 is called a root of the polynomial f .
A 1n a 2n ⎟ ⎟ .. ⎟ . ⎠ 0 ··· a nn Let us suppose that in the process of doing this, we have completed t elementary operations of type I and some number of operations of type II. Since operations of type II do not change the determinant, and an operation of type I multiplies the determinant by −1, we have |A| = (−1)t |A|. We shall now show that |A| = a 11 a 22 · · · a nn . 20) |A| = (−1)t a 11 a 22 · · · a nn . 21) Then This is a formula for calculating |A|. 20) by induction on n. 22) in which the determinant a 22 0 A = .