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Unformatted text preview: M340L: Matrices and Matrix Calculations Unique Number: 57160 PRACTICE FIRST TEST, February, 2006- Calculators are authorized. No other document is authorized neither book nor hand written notes.- Show all of your work.- Justify every answer, either by a calculation or by using a theorem. Results without justification will not be taken care of even if they are correct.- Use freely both sides of the sheets if necessary.- Put your name on any additional or unstapled sheet.- Good Luck. I give first the questions without answers and then again with the answers. In the real test you will have a blank space below each question to answer it. Problem I: Let us consider the matrix: A = 1 2 1 2 2 4 1 3- 1 1 2- 2- 1- 3 3 6- 2 1- 5 . and the linear mapping T A with standard matrix A . 1) What is the source or domain of T A ?(5pts.) What is the target or codomain of T A ? (5pts.) 2) Give an echelon form for A (5pts.). Give the reduced echelon form of A (5pts.). Take the time to do this properly. To get the full grade you must show all the computations and describe the row operation at each step ( e.g. R i R i + cR j ). 3) 3.1 Are the columns of A linearly independent? (5pts) Which columns of A are pivot columns? (5pts.) 3.2 What is a one-to-one mapping (give a definition that stands either the mapping is linear or not)?(5pts) 3.3 Is the linear mapping T A one-to-one (or injective)? Why or why not? (5pts) 3.4 Solve the homogeneous linear system A x = and give the solution in parametric vector form (5pts + 2pts). 4) 4.1 Do the columns of A span R 4 ? Why or why not ? (5pts) 4.2 What is an onto mapping (give a definition that stands either the mapping is linear or not)(5pts). Is the linear mapping T A onto (or surjective)? (5pts) 4.2 What is an onto mapping (give a definition that stands either the mapping is linear or not)(5pts). Is the linear mapping T A onto (or surjective)? (5pts) 1 4.3 Check the consistency and solve (when it is possible)the systems A x = b i , i = 1 , 2, for b 1 = 4 7 1 7 and b 2 = 4 7 (3pts each) 4.4 Would you say that b 1 belongs to the subspace spanned by the columns of A ? Why or why not?(5pts) What about b 2 and why is it so? (2pts) Problem 2. Let us consider the matrix A = 1 2 3 2 5 3 1 0 8 . Knowing that A is invertible, without any computation determine: 1) the solution set of the homogeneous equation...
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