10) EXPLAIN why the columns of matrix A are linearly independent if and only if

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10) EXPLAIN why the columns of matrix A are linearly independent if and only if the equation Ax = 0 has only a trivial solution. (11) What can you say about a matrix that has a pivot in each column? What can you say about a matrix that has a pivot in each row? (12) Explain why three independent vectors in R^3 span R^3 . Do the columns of the matrix A given in problem (2) above span R^3 ? Yes. (13) Which of the following subsets of R 4 are subspaces? Give a brief reason. a. All vectors b = (b1; b2; b3; b4) satisfying b1 = b2b3. no b. All linear combinations of v = (1;2;3;4) and w = (1; 1;1; 1). yes c. All vectors b = (x1; x2; x3; x4) satisfying x1 2x2 + 3x3 4x4 = 0. yes d. All vectors b = (x1; x2; x3; x4) satisfying x1 2x2 + 3x3 4x4  0. no (14) Let W1 be the subset of Pn consisting of p(t) so that p(0)p(1) = 0 and let W2 be the subset of Pn consisting of p(t) so that p(2) = 0. Which of the two (if any) is a subspace of Pn. W2

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300 points is too less for this question .... :-/

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