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

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

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