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lecture slides for introduction to machine leraning
Typology: Lecture notes
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Vectors x 1 , x 2 , · · · , x (^) n are linearly independent if no combination gives c 1 x 1 + c 2 x 2 + · · · + c (^) n xn = 0 except for c 1 = c 2 = · · · = c (^) n = 0. For real vectors, linear independence is equivalent to non-collinearity (algebraic vs. geometrical meaning) 0 vector cannot be independent with any non-zero vector v , because we always have c 0 + 0v = 0 for c 6 = 0. Column vectors v 1 , v 2 , · · · , v (^) n of matrix A (^) m⇥n are Linearly independent if N(A) only includes 0 ) r = n m, no free variables Linearly dependent if 9 vector x 6 = 0 s.t. Ax = 0 ) r < n, there are n r free variables Vectors v 1 , v 2 , · · · , v (^) n span a space if that space consists of ALL linear combinations of those vectors. e.g., two vectors cannot span R 3 e.g., four non-coplanar vectors span R 3 , but they must be linearly dependent!
List them as the column vectors of A and perform Gaussian elimination, then count the number of pivots in the Echelon form.
In general, n vectors form a valid basis of Rn^ if the n ⇥ n matrix they are involved in is invertible (non-singular). There can be infinite number of di↵erent bases Common: number of vectors (rank) ) Dimension of the space
(^5) r = dim C (A) = 2
The first 2 columns are pivot columns, so {v 1 , v 2 } is a basis Since v 3 = v 1 + v 2 , {v 1 , v 3 } is also a basis Since v 4 = 2v 1 , {v 1 , v 4 } is NOT a basis
We have already defined Column space C (A) ⇢ R m^ : dim C (A) = r Null space N(A) ⇢ R n^ : Kernel of A, dim N(A) = n r ”Nullity” Similarly, by switching the roles of columns and rows, Row space C (A T^ ) ⇢ R n Left null space N(A T^ ) = {x|A T^ x = 0 or x T^ A = 0} ⇢ R m
Subspace C (A) C (AT^ ) N(A) N(AT^ ) Dimension r r n r m r Basis Pivot col. Pivot col. Setting each Do the same of A of AT^ free variables 1 for AT^ as and solve Ux = 0 solving N(A)