Math416_HW9_sol - Math 416 Abstract Linear Algebra Fall...

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Unformatted text preview: Math 416 - Abstract Linear Algebra Fall 2011, section E1 Homework 9 solutions Section 5.2 2.2. (2 pts) Let x ∈ R n and consider the equivalent conditions: x ∈ (Col A T ) ⊥ = (Row A ) ⊥ ⇔ x ⊥ row i of A for all i = 1 ,...,m ⇔ (row 1 of A,x ) . . . (row m of A,x ) = ~ ∈ R m ⇔ Ax = ~ ⇔ x ∈ Null A. Therefore we have (Row A ) ⊥ = Null A . In words: the null space is the orthogonal complement of the row space. Applying that result to the matrix A T instead of A , we obtain (Col A ) ⊥ = Null A T . In words: the left null space is the orthogonal complement of the column space. 2.3. b. Because v 1 ,...,v n is an orthonormal basis, the coordinates of x in that basis are given by x = n X i =1 ( x,v i ) v i . Using part (a), with α i = ( x,v i ) and β i = ( y,v i ), be obtain ( x,y ) = n X i =1 α i β i = n X i =1 ( x,v i ) ( y,v i ) . c. Assuming v 1 ,...,v n is an orthogonal basis, the coordinates of x in that basis are given by x = n X i =1 ( x,v i ) ( v i ,v i ) v i . 1 Using this, we compute the inner product of any two vectors ( x,y ) = ( n X i =1 ( x,v i ) ( v i ,v i ) v i , n X j =1 ( y,v j ) ( v j ,v j ) v j ) = X i,j ( x,v i ) ( v i ,v i ) ( y,v j ) ( v j ,v j ) ( v i ,v j ) by (anti)linearity and ( v j ,v j ) ∈ R = n X i =1 ( x,v i ) ( v i ,v i ) ( y,v i ) ( v i ,v i ) ( v i ,v i ) since the v i are orthogonal = n X i =1 ( x,v i ) ( y,v i ) ( v i ,v i ) . Remark: This is consistent with part (b): n X i =1 ( x,v i ) ( y,v i ) ( v i ,v i ) = n X i =1 ( x,v i ) ( y,v i ) k v i k 2 = n X i =1 ( x, v i k v i k ) ( y, v i k v i k ) and v 1 k v 1 k ,..., v n k v n k is an orthonormal basis. 2.5. (1 pt check) Because v 1 ⊥ v 2 , the orthogonal projection of v = 1 1 1 1 onto Span { v 1 ,v 2 } is Proj { v 1 ,v 2 } ( v ) = ( v,v 1 ) ( v 1 ,v 1 ) v 1 + ( v,v 2 ) ( v 2 ,v 2 ) v 2 = 6 12 v 1 + 2 6 v 2 = 1 6 3 1 3 1 1...
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This note was uploaded on 12/16/2011 for the course MATH 416 taught by Professor Frankland during the Fall '11 term at University of Illinois at Urbana–Champaign.

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Math416_HW9_sol - Math 416 Abstract Linear Algebra Fall...

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