linalg - Brief review of linear algebra plus some...

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Brief review of linear algebra, plus some additional facts 1. R n x = x 1 . . . x n ± ± ± ± ± ± ± x i real . 2. A linear combination of vectors x 1 ··· , x k in R n is a vector of the form x = c 1 x 1 + + c k x k where c 1 , ..., c k are scalars. 3. A set of vectors { x 1 , ..., x k } is linearly dependent if some nonzero linear combination of them yields the zero vector; otherwise it is linearly independent . 4. A subset V R n is a subspace of R n if x , y V = c x + d y V , i.e., if V is closed under linear combination. 5. A set of vectors { x 1 , ..., x k } in a subspace V is a basis for V if each x V can be expressed uniquely as a linear combination of x 1 , ... x k . The dimension of V , denoted dim( V ), is the number of vectors in a basis for V . Result : A set of vectors in R n is a basis for R n iff there are n of them in all and they are linearly independent. 6. span( x 1 , ..., x k ) is the set of all linear combinations of x 1 , ..., x k . 7. R m,n A = a 1 , 1 a 1 ,n . . . . . . a m, 1 a m,n ± ± ± ± ± ± ± a i,j real . 8. For A R m,n ... column space( A ) span(columns of A ). row space( A ) span(rows of A ). range( A ) ≡{ y R m | y = A x for some x R n } . null( A ) x R n | A x =0 } . rank( A ) dim(column space( A )) (= dim(row space ( A )) 9. The transpose of A R m,n is a matrix A T R n,m whose i, j th entry is a j,i . 10. A R n,n is nonsingular if there exists a matrix A - 1 R n,n such that AA - 1 = I , where I is the n × n identity matrix .I f A - 1 exists, it is also the case that A - 1 A = I . 11. Result :( AB ) T = B T A T ;( AB ) - 1 = B - 1 A - 1 (assuming A and B are nonsingu- lar). 12. The inner product (or dot product , or scalar product ) of two vectors x , y R n is: ( x , y ) n X i =1 x i y i (= x T y ) . Typeset by A M S -T E X
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13. Two vectors x , y R n are orthogonal if ( x , y )=0. 14. { x 1 ,...,x k } is an orthogonal set of vectors if ( x i , x j )=0for i 6 = j . 15. { x 1 ,..., x k } is an orthonormal set if ( x i , x j )= δ i,j ( 1 ,i = j 0 6 = j. Result : An orthonormal set of vectors is linearly independent. In fact, if { x 1 , ..., x k } is an orthornormal set and x = c 1 x 1 + ··· + c k x k ,then c i =( x , x i ) =1 , ..., k . 16. An orthogonal matrix is a real matrix whose transpose is its inverse. Example: ± cos θ sin θ - sin θ cos θ ² (rotation through angle θ in R 2 ) Results : (i) The product of orthogonal matrices is orthogonal. Proof : Let Q 1 ,Q 2 be orthogonal. Then ( Q 1 Q 2 ) T ( Q 1 Q 2 Q T 2 Q T 1 Q 1 Q 2 = Q T 2 IQ 2 = Q T 2 Q 2 = I. (ii) A real matrix is orthogonal iff its columns (rows) comprise an orthonormal set of vectors. Proof : Note that matrix multiplication can be defined in terms of inner products as follows: ( AB ) i,j = (row i of A, column j of B ). Thus Q is orthogonal ⇐⇒ Q T Q = I ( Q T Q ) i,j = δ i,j (row i of Q T , column j of Q δ i,j ( q ( i ) ,q ( j ) δ i,j, where q ( j ) is the j th column of Q .
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This note was uploaded on 10/06/2010 for the course CS CS 510 taught by Professor Richter during the Fall '10 term at Rutgers.

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linalg - Brief review of linear algebra plus some...

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