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Simple introduction to Linear Algebra i) C. W. Curtis, Linear Algebra: An Introductory Approach, Springer ii) G. Schay, A Concise Introduction to Linear Algebra, Springer Linear programming i) J. Matousek, B. Gärtner, Understanding and Using Linear Programming, Springer ii) R. J. Vanderbei, Linear Programming: Foundations and Extensions, Springer
Examples: R2, Rn. Polynomials of degree at most two: P2, p(x) = a0 + a1x + a2x2. The straight line: f(x; y) 2 R2 x + y = 0g is a linear space. It is a. linear subspace of R2. The straight line: f(x; y) 2 R2 x + y = 1g is not a linear space since it does not have the zero vector. The upper half plane f(x; y) 2 2.
V = L p([a,b]), ∥f∥= Z b a |f(x)|pdx 1/p. Example The space V = l pof all sequences {a k}∞ k=1 such that ∥{a}∥ p:= X∞ k=1 |a k|p! 1/p <∞. In particular, l 1 is the space of all absolutely convergent sequences as ∥{a}∥ 1:= X∞ k=1 |a k|<∞. Example The space V = l ∞of all sequences {a k}∞ k=1 such that ∥{a}∥ ∞:= sup ...
FUNDAMENTALS OF LINEAR ALGEBRA James B. Carrell carrell@math.ubc.ca (July, 2005)
We make linear approximations to real life problems, and reduce the problems to systems of linear equations where we can then use the techniques from Linear Algebra to solve for approximate solutions.
Part 1 : Basic Ideas of Linear Algebra. 1.1 Linear Combinations of Vectors. 1.2 Dot Products v · w and Lengths || v || and Angles θ. 1.3 Matrices Multiplying Vectors : A times x. 1.4 Column Space and Row Space of A. 1.5 Dependent and Independent Columns. 1.6 Matrix-Matrix Multiplication AB. 1.7 Factoring A into CR : Column rank = r = Row rank.
Examples. The field k itself is a k-vector space, with its own multiplication as scalar multiplication. A trivial group (with one element) is always a k-vector space (with the only possible scalar multiplication). 1 Suppose V is a k-vector space. Prove that. 0 · v = 0, for all v ∈ V. (The zero in the left-hand side is the field k.
