New PDF release: Linear Algebra in 25 Lectures

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By Denton, Waldron

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The second computation is slightly easier:     x1 + x3 − x4 y1 + y3 − y4 f (X) = M X = x2 − x3 + x4  and f (Y ) = M Y = y2 − y3 + y4  , 0 0 (using our result above). Adding these gives   x1 + x3 − x4 + y1 + y3 − y4   f (X) + f (Y ) = x2 − x3 + x4 + y2 − y3 + y4  . 0 Next we perform the first computation beginning with:   x1 + y1 x2 + y2   X +Y = x3 + y3  , x4 + y4 from which we calculate  x1 + y2 + x3 + y3 − (x4 + y4 )    f (X + Y ) = x2 + y2 − (x3 + y3 ) + x4 + y4  .

So we make the following definition. Definition A vector space (over R) is a set V with two operations + and · satisfying the following properties for all u, v ∈ V and c, d ∈ R: (+i) (Additive Closure) u + v ∈ V . ) (+ii) (Additive Commutativity) u + v = v + u. ) (+iv) (Zero) There is a special vector 0V ∈ V such that u + 0V = u for all u in V . (+v) (Additive Inverse) For every u ∈ V there exists w ∈ V such that u + w = 0V . (· i) (Multiplicative Closure) c · v ∈ V . ) (· ii) (Distributivity) (c+d)·v = c·v +d·v.

We can think of these functions as infinite sequences: f (0) is the first term, f (1) is the second term, and so on. Then for example the function f (n) = n3 would look like this: f = {0, 1, 8, 27, . . , n3 , . }. Thinking this way, V is the space of all infinite sequences. Let’s check some axioms. (+i) (Additive Closure) f1 (n) + f2 (n) is indeed a function N → R, since the sum of two real numbers is a real number. (+iv) (Zero) We need to propose a zero vector. The constant zero function g(n) = 0 works because then f (n) + g(n) = f (n) + 0 = f (n).

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Linear Algebra in 25 Lectures by Denton, Waldron

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