Math416_HW9_sol

# Math416_HW9_sol - Math 416 Abstract Linear Algebra Fall...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

### Page1 / 9

Math416_HW9_sol - Math 416 Abstract Linear Algebra Fall...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online