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STATS 100C: Linear Models


Problem Set 1


Problem 1.1

Consider the matrices

(a) Determine the rank of the two matrices A and B.

(b) Determine the rank of the two matrices AB and BA.

(c) Determine the kernel of A and B.

(d) Find a basis for Im(A) = column space of A.

(e) Express the orthogonal complement of Im(B), that is, Im(B)丄,as the span of a set of vectors.


Problem 1.2

Find the projection of vector x e R3 onto vector a e R3 where


Problem 1.3

Consider the following the matrix:

(a) Find the spectral decomposition of A, that is write A = UAUT where U is orthogonal and A is diagonal.

(b) Use the spectral decomposition, to determine the rank of A.

(c) Is A positive semidefinite (PSD)?