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Assignment1

# Assignment1 - of A and b(Hint Remember that the minimum...

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data Mining - Spring 2011 Home assignment 1 Lots of the data mining methodology is based on numerical linear algebra. The first assign- ment is supposed to be done without computers or calculators. 1. Let a (1) , . . . , a ( k ) R n be linearly independent vectors, k < n . In class, the orthogonal projection P : R n span { a (1) , . . . , a ( k ) } was given in the special case when the vectors a ( j ) for an orthonormal set. General- ize the result in the case when the vectors are not necessarily orthonormal. The two fundamental properties of the projection matrix are that for any x R n , x P x a ( j ) , 1 j k, and P x = x for x span { a (1) , . . . , a ( k ) } . In particular, check that your solution satisfies P 2 = P , a fundamental property of a projection matrix. 2. Let A R n × k , k < n , and assume that the columns of A are linearly independent. Consider the least squares problem (LSQ): find an x R k so that x = argmin A x b , where b R n . Use the solution of the previous problem to find this solution in terms
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Unformatted text preview: of A and b . (Hint: Remember that the minimum distance of a vector from a linear subspace is along the line that is along the line that is perpendicular to that subspace.) 3. Consider the matrix A =   1 0 1 − 2 1 0 1 2 5   . (a) What is the rank of the matrix A ? (b) Write the matrix in the form A = k ∑ j =1 u ( j )( v ( j ) ) T , where k is the rank of A . (c) Write explicitly the projection matrix onto the span of the columns of A . 4. Consider the linear mapping R 3 → R 3 , x 7→ v × x, where v is a ﬁxed vector in R 3 and “ × ” is the cross product in R 3 . (a) Find a matrix A ∈ R 3 × 3 representing the linear operation. (b) Describe the null space and the range of this matrix. What is the range of the matrix?...
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