7. By whatever means you see fit, determine if the following sets of vectors for
ID: 3185476 • Letter: 7
Question
7. By whatever means you see fit, determine if the following sets of vectors form a basis for the indicated vector space. Justify your answers. (a) (S points) ls D-RH?] (b) (5 points) Is B- , | [:] } a basis fr R27 | | i | } a basis for R17 0 '3 f a basis for R2? (c) (5 points) Is C2,-1a basis for R3? 2 8. Answer each of the following. Justify your answers. (a) (5 points) Suppose A E M5,7(R). Are the columns of A linearly dependent? (b) (5 points) Suppose B E M4,6(R) and B O. What are all the possible values of nullity(B)? (c) (5 points) If C e M8,5(R), are the rows of C linearly independent?Explanation / Answer
7(a). We know that the dimension of R2 is 2 so that any set of more than 2 vectors in R2 is linearly dependent. Hence the set D is not a basis for R2 .
(b). Let A =
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0
-1
-1
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The RREF of A is
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It implies that (-1,0,1,0)T = -1(1,-1,0,0)T-1(0,1,0,-1)T-(0,0,-1,1)T. Thus, the set B is linearly dependent. Hence B is not a basis for R4.
( c). We know that the dimension of R3 is 3 so that any set of less than 3 vectors in R3 cannot be a basis for R3. Hence C is not a basis for R3.
8. (a) Since A is a 5x7 matrix, it has 7 columns each of which is a 4-vector. Further, dim(R4 ) = 4 so that the columns of A, being 7 in number, are linearly dependent.
(b).B is a non-zero 4x6 matrix. Hence, the rank of B is 1,2,3 or 4. Thus, by the rank-nullity theorem, the nullity of B is 5,4,3 or 2.
( c). C is a 8 x 5 matrix. Since the column rank and the row rank of a matrix are equal, hence the maximum possible rank of C is 5. Therefore, the rows of C, being 8 in number,are linearly dependent.
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