Q1.(20 points, 4 points each)
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Q2.(25 points, 5 points each) Fill in the blanks
(1)Let with , and Then
(2) Let be an matrix with and let be the identity matrix. Then
(3)Letwith Then .
(4)Let be a nonzero 3-dimensional real column vector in with , and be the identity matrix. Then rank .
(5) Let Then the least squares solution to is
Q3 (15 points) Let , and
(a) By applying row operations, determine for which values of is the matrix invertible?
(b) Find the values of such that the nullspace of , has dimension 1?
(c) Let .Write down the matrix inverse of .
Q4.(10points) Let
Find an LU factorization of A.
Q5.(10 points) Consider the following system ofline are quations:
Note that the above system has four variables Suppose another homogeneous system of linear equations has special solutions
Find the common nonzero solutions of systems and .
Q6.(8 points) Let be the vector space consisting of all real matrices. Let .
(a) Show that is a basis for .
(b) Show that : is a linear transformation.
(c)Find the matrix representation of with respect to the ordered basis .
Q7.(6 points) Let A,B be two matrices. Prove
(a)
(b)
Q8.(6 points) Let be an matrix with rank Show that there exist an matrix and an matrix such that and both and have rank