Lab3Sol - NATIONAL UNIVERSITY OF SINGAPORE Department of...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA1101R Laboratory 3 Semester I 2013/14 Linaer Combinations, Linear Span, Linear Independence, Transformation Matrices In this laboratory session, we will consolidate what we have learnt in lectures in- volving concepts of linear combinations/span/independence. We will also use the specially written matlab functions trans2x2 and trans3x3 to visualize the geomet- rical effect of certain matrices, when these matrices are multiplied to plane figures in R 2 . Activity 1 Enter the following matrix (carefully!) in the matlab command window. A = 0 2 1 - 3 2 - 2 2 1 - 1 0 5 - 2 4 0 1 3 4 5 2 9 0 2 1 2 7 - 1 0 2 . (i) Write down the reduced row-echelon form of A . 1 0 0 2 - 1 0 2 0 1 0 - 3 1 0 2 0 0 1 3 0 0 - 2 0 0 0 0 0 1 0 (ii) Let a 1 = 0 1 1 2 , a 2 = 2 - 1 3 1 , a 3 = 1 0 4 2 , a 4 = - 3 5 5 7 , a 5 = 2 - 2 2 - 1 , a 6 = - 2 4 9 0 , a 7 = 2 0 0 2 . 1
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(So a i is the i -th column of A .) By just using the answer from (i), without performing any further computation in matlab , you should be able to answer the following questions. (a) What will be the reduced row-echelon form of the matrix ( a 1 a 2 a 3 | 0 )? 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 (b) Is { a 1 , a 2 , a 3 } a linearly independent set? Why? Yes, since c 1 a 1 + c 2 a 2 + c 3 a 3 = 0 has only the trivial solution, as seen from part (b). (c) Is span { a 1 , a 2 , a 3 , a 4 } = R 4 ? Why? No, since the reduced row echelon form of ( a 1 a 2 a 3 a 4 ) has a row of zeros.
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