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Clemson University ECE 8010 ECE8010_HW4_solution.

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ECE 8010 Homework #4 Solutions 1. (Exercise) The elementary row operations used to find reduced row echelon form of a matrix are (i) scaling row i by  , (ii) swapping row i and row j, and (iii) replacing row i with t ...

ECE 8010 Homework #4 Solutions 1. (Exercise) The elementary row operations used to find reduced row echelon form of a matrix are (i) scaling row i by  , (ii) swapping row i and row j, and (iii) replacing row i with the sum of row i and row j. a. Performing an elementary row operation on a matrix A is equivalent to multiplying A on the left by an appropriate square matrix M , i.e. MA. For each of the elementary row operations, describe the form of the matrix M and check that the matrix is full rank. b. Similarly, elementary column operations are equivalent to right multiplication by an appropriate square matrix M , i.e. AM . For each of the elementary column operation, describe the form of the matrix M and check that the matrix is full-rank. 2. (Exercise) Let :    n n  , :    n m  , :    m m  be linear transformations such that ( )   n and ( )   m . Use the Sylvester Inequality to show that   ( ) ( )    . 3. Let : n n A    , : m n B    , : m m C    be linear transformations. (The notation A B C , , will be used for the transformation as well as the corresponding matrix representation of the transformation when the standard bases are used for the domain and codomain.) Let A and C be full rank. a. Show that   ( ) ( ) AB B  b. Find a counterexample to demonstrate that in general   ( ) ( ) BC B  c. Show that   ( ) ( ) BC B  d. Find a counterexample to demonstrate that in general   ( ) ( ) AB B  e. (Exercise) Can you find a condition related to m , n , and ( ) B that will guarantee that the columns B form a basis for ( ) B ? Note that 3a is a stronger statement than   ( ) ( ) AB B  . While   ( ) ( ) AB B  implies   ( ) ( ) AB B  , the converse is not true. Similarly for 4b,   ( ) ( ) BC B  implies   ( ) ( ) BC B  , but the converse is not true. 4. (Exercise) True/False: (Provide a proof if true, or a counterexample if false.) a. A full rank linear transformation is always onto. b. A full rank linear transformation is always 1:1. c. A square, full rank linear transformation is always 1:1

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