By Stephen Francis Andrilli; David Hecker
"Elementary Linear Algebra, Fourth version deals computational concepts and primary theoretical effects significant to a primary direction in linear algebra. The textual content makes a steady and soft transition from computational effects concerning vectors, matrices, and platforms of linear equations to the final idea of summary vector areas. The textual content additionally offers a complete variety of sensible purposes, which might be coated in a versatile manner." "This textual content assists scholars to achieve a degree of summary pondering crucial for additional mathematical examine. The authors provide early, in depth consciousness to the abilities wanted for a scholar to develop into ok with the examining and writing of mathematical proofs. This emphasis is sustained in the course of the textual content with cautious statements of theorems and their proofs, and proof-related exercises."--BOOK JACKET. Read more...
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Extra info for Elementary linear algebra
Zn ] be any vectors in Rn , and let c and d be any real numbers (scalars). Let 0 represent the zero vector in Rn . Then (1) (2) (3) (4) (5) (6) (7) (8) xϩy ϭ y ϩx x ϩ (y ϩ z) ϭ (x ϩ y) ϩ z 0ϩx ϭ xϩ0 ϭ x x ϩ (Ϫx) ϭ (Ϫx) ϩ x ϭ 0 c(x ϩ y) ϭ cx ϩ cy (c ϩ d)x ϭ cx ϩ dx (cd)x ϭ c(dx) 1x ϭ x Commutative Law of Addition Associative Law of Addition Existence of Identity Element for Addition Existence of Inverse Elements for Addition Distributive Laws of Scalar Multiplication over Addition Associativity of Scalar Multiplication Identity Property for Scalar Multiplication In part (3), the vector 0 is called an identity element for addition because 0 does not change the identity of any vector to which it is added.
Two vectors are orthogonal if and only if their dot product is zero. ■ Two vectors are parallel if and only if their dot product is either equal to or opposite the product of their lengths. 28 CHAPTER 1 Vectors and Matrices ■ The projection of a vector b onto a vector a is found by multiplying a by the scalar (a · b)/||a||2 . ■ Any vector can be expressed as the sum of two component vectors such that one (if nonzero) is parallel to a given vector a, and the other is orthogonal to a. ■ The work accomplished by a vector force is equal to the dot product of the vector force and the vector displacement.
If A Then B” Proofs Frequently, a theorem is given in the form “If A then B,” where A and B represent statements. ” The entire“If A then B”statement is called an implication; A alone is the premise, and B is the conclusion. The meaning of “If A then B” is that, whenever A is true, B is true as well. Thus, the implication “If x ϭ 0, then x ϭ 0” means that, if we know x ϭ 0 for some particular vector x in Rn , then we can conclude that x is the zero vector. 4 Therefore,to prove“If A then B,”we assume A is true and try to prove B is also true.
Elementary linear algebra by Stephen Francis Andrilli; David Hecker