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Linear Algebra

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4 2 1 0  4 6 1 3  . 1. Let A    8 16 3 4     20 10 4 3

(a) Find an LU-factorization of A i.e. use row operations to find U, an upper triangular matrix equivalent to A and L, a lower triangular matrix such that A  LU . (b) Find the determinant of A.  3 1 3  1  4 2 0  and b   1  . 2. Let A      2 2 1 4    

(a) Find the determinant of A. (b) Solve the linear system Ax  b by the Cramer’s rule. a  3. Let V be the set of all 2 1 real matrices v    , where a and b are integers such b   3 8   1  1 that a  b is even. Examples of matrices in V are   ,   ,   , and   .  5   2  7   1 Let the operation  be standard addition of matrices and the operation be standard scalar multiplication of matrices on V. Is V a vector space? Justify your answer.

4. The following set together with the given operations is not a vector space. List the properties in the definition of a vector space that fail to hold. a  V is the set of all 2 1 real matrices v    , with operation  be standard matrix b  addition and the operation be scalar multiplication

c

 a   c ( a  b)  b   c(a  b)  , for any real number c.    

Note: This assignment must be submitted to your respective tutor (or deposit your assignment in the prescribed MAT111 pigeon-hole on the ground floor of Building G31) on or before Monday, 1 April 2013. Late submission will not be entertained.