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Problem 4: (10 points) Which of the following statements is FALSE? {εζ₯} B. If the columns of a 6x4 matrix A are linearly independent, then A has 4 pivot columns. {8.8.8} D. The columns of any 5 x 4 matrix are linearly dependent. E. If Az = 0 has only the trivial solution, then the columns of A are linearly independent. A. The set C. The set is linearly dependent. is a basis for RΒ³.
Problem 3: (10 points) Let A be an n x n invertible matrix, which of the following state- ments is/are always true. (i) Az = 0 has at least one nontrivial solution. (ii) A is row equivalent to In (iii) The columns of A form a basis for R". (iv) Any b in R" is in the column space of A. A. (i) only. B. (ii) and (iii) only. C. (i), (ii) and (iii) only. D. (ii), (iii) and (iv) only. E. All the statements are true. 5/13 βββ ...
Problem 5: (10 points) Let A be a 5 x 7 matrix, and let B denote the reduced row echelon form of A. Which of the following statements is TRUE? A. If the number of nonzero rows of B equals 4, then the number of non-zero columns of B equals 4. B. If the columns 1, 2, 5, 6 of B have leading I's, then the columns 1, 2, 5, 6 of A span RΒΉ. C. If the number of nonzero rows of B equals 4, then one can pick 4 columns of A that form a basis of the column space of A. D. If B has 4 leading 1's, the dimension of the null space of A is 3 and therefore Nul(A) = RΒ³. E. None of the above.