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Unformatted text preview: Linear Algebra Jim Hefferon ( 2 1 ) ( 1 3 ) ﬂ ﬂ ﬂ ﬂ 1 2 3 1 ﬂ ﬂ ﬂ ﬂ ( 2 1 ) x 1 · ( 1 3 ) ﬂ ﬂ ﬂ ﬂ x · 1 2 x · 3 1 ﬂ ﬂ ﬂ ﬂ ( 2 1 ) ( 6 8 ) ﬂ ﬂ ﬂ ﬂ 6 2 8 1 ﬂ ﬂ ﬂ ﬂ Notation R real numbers N natural numbers: { , 1 , 2 ,... } C complex numbers { ... ﬂ ﬂ ... } set of . .. such that . .. h ... i sequence; like a set but order matters V,W,U vector spaces ~v, ~w vectors ~ 0, ~ V zero vector, zero vector of V B,D bases E n = h ~e 1 , ..., ~e n i standard basis for R n ~ β, ~ δ basis vectors Rep B ( ~v ) matrix representing the vector P n set of nth degree polynomials M n × m set of n × m matrices [ S ] span of the set S M ⊕ N direct sum of subspaces V ∼ = W isomorphic spaces h,g homomorphisms, linear maps H,G matrices t,s transformations; maps from a space to itself T,S square matrices Rep B,D ( h ) matrix representing the map h h i,j matrix entry from row i , column j  T  determinant of the matrix T R ( h ) , N ( h ) rangespace and nullspace of the map h R ∞ ( h ) , N ∞ ( h ) generalized rangespace and nullspace Lower case Greek alphabet name character name character name character alpha α iota ι rho ρ beta β kappa κ sigma σ gamma γ lambda λ tau τ delta δ mu μ upsilon υ epsilon ² nu ν phi φ zeta ζ xi ξ chi χ eta η omicron o psi ψ theta θ pi π omega ω Cover. This is Cramer’s Rule for the system x + 2 y = 6, 3 x + y = 8. The size of the ﬁrst box is the determinant shown (the absolute value of the size is the area). The size of the second box is x times that, and equals the size of the ﬁnal box. Hence, x is the ﬁnal determinant divided by the ﬁrst determinant. Preface In most mathematics programs linear algebra comes in the ﬁrst or second year, following or along with at least one course in calculus. While the location of this course is stable, lately the content has been under discussion. Some instructors have experimented with varying the traditional topics and others have tried courses focused on applications or on computers. Despite this healthy debate, most instructors are still convinced, I think, that the right core material is vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. Applications and code have a part to play, but the themes of the course should remain unchanged. Not that all is ﬁne with the traditional course. Many of us believe that the standard text type could do with a change. Introductory texts have traditionally started with extensive computations of linear reduction, matrix multiplication, and determinants, which take up half of the course. Then, when vector spaces and linear maps ﬁnally appear and deﬁnitions and proofs start, the nature of the course takes a sudden turn. The computation drill was there in the past because, as future practitioners, students needed to be fast and accurate. But that has changed. Being a whiz at 5 × 5 determinants just isn’t important anymore....
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 Spring '10
 Dr.McGrawHill
 Linear Algebra, Algebra, Linear Systems, Vector Space, The Land

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