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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 onetoone mapping (give a definition that stands either the mapping is linear or not)?(5pts) 3.3 Is the linear mapping T A onetoone (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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 Spring '11
 SECKIN
 Matrices

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