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Unformatted text preview: Answers to Exercises 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 , R + , R n real numbers, reals greater than 0, ntuples of reals N natural numbers: { , 1 , 2 , . . . } C complex numbers { . . . ﬂ ﬂ . . . } set of . . . such that . . . ( a .. b ), [ a .. b ] interval (open or closed) of reals between a and b . . . sequence; like a set but order matters V, W, U vector spaces v, w vectors 0, 0 V zero vector, zero vector of V B, D bases E n = e 1 , . . . , e n 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 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 1 + 2 x 2 = 6, 3 x 1 + x 2 = 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 1 times that, and equals the size of the ﬁnal box. Hence, x 1 is the ﬁnal determinant divided by the ﬁrst determinant. These are answers to the exercises in Linear Algebra by J. Hefferon. Corrections or comments are very welcome, email to jimjoshua.smcvt.edu An answer labeled here as, for instance, One.II.3.4, matches the question numbered 4 from the ﬁrst chapter, second section, and third subsection. The Topics are numbered separately. Contents Chapter One: Linear Systems 4 Subsection One.I.1: Gauss’ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Subsection One.I.2: Describing the Solution Set . . . . . . . . . . . . . . . . . . . . . . . 10 Subsection One.I.3: General = Particular + Homogeneous . . . . . . . . . . . . . . . . . 14 Subsection One.II.1: Vectors in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Subsection One.II.2: Length and Angle Measures . . . . . . . . . . . . . . . . . . . . . . 20 Subsection One.III.1: GaussJordan Reduction . . . . . . . . . . . . . . . . . . . . . . . 25 Subsection One.III.2: Row Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Topic: Computer Algebra Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Topic: InputOutput Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
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 Spring '06
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 Linear Algebra, Algebra, Differential Calculus, Vector Space, subsection

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