{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW3_Sol

# HW3_Sol - Math 104 Fall 2009 Homework 3 solution set We use...

This preview shows pages 1–2. Sign up to view the full content.

Math 104 Fall 2009 Homework 3: solution set We use the notations below to ease readability. Matrices are bold capital, vectors are bold lowercase and scalars or entries are not bold. For instance, A is a matrix and a ij (sometimes, A ( i, j )) its ( i, j )th entry. Likewise x is a vector and x j its j th component. The linear span generated by a group of vectors v 1 , v 2 , . . . , v n is denoted by span( v 1 , v 2 , ..., v n ). Problem 1 null space; left null space. Problem 2 Set v 1 = 1 1 1 and v 2 = 1 0 1 . We proceed by finding an orthonomal basis of span( v 1 , v 2 ). Let u 1 = v 1 - v 2 = 0 1 0 and u 2 = v 2 k v 2 k = 1 2 1 0 1 . Then u 1 , u 2 span( v 1 , v 2 ), k u 1 k = k u 2 k = 1 and u * 1 u 2 = 0. Since dim(span( v 1 , v 2 )) = 2, { u 1 , u 2 } is an orthonormal basis of span( v 1 , v 2 ). The orthogonal projection onto span( u 1 , u 2 ) is P = u 1 u * 1 + u 2 u * 2 = 1 2 0 1 2 0 1 0 1 2 0 1 2 . Problem 3 (a) Suppose λ is an eigenvalue of A . Then there is a nonzero vector x C m obeying Ax = λ x , and x * Ax = x * λ x = λ k x k 2 . (1)

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.

{[ snackBarMessage ]}

### Page1 / 2

HW3_Sol - Math 104 Fall 2009 Homework 3 solution set We use...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online