Linear Algebra - Chaps 1-2, true false questions
1-a) every matrix is row equivalent to a unique matrix in echelon form.
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1-b) any system of n linear equations in n variables has at most n solutions.
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1-d) if a system of linear equations has two different solutions, it must have infinitely many solutions.
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1-g) If A is an m x n matrix and the equation Ax=b is consistent for every b in Rm, then A has m piviot columns.
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1-h) If an augmented matrix [A b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax=b is consistent.
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1-j) The equation Ax=0 has the trivial solution if and only if there are no free variables.
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1-l) IF an m x n matrix A has a pivot position in every row, then the equation Ax has a unique solution for each b in Rm.
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1-q) If none of the vectors in the set S = {v1, v2, v3} in R3 is a multiple of one of the other vectors, then S is linearly independent.
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1-s) in some cases, it is possible for four vectors to span R5.
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1-t) If u and v are in Rm, then -u is in Span{u,v}
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1-u) if u, v, and w are nonzero vectors in R2, then w is a linear combination of u and v.
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1-x) a linear transformation is a function.
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1-z) IF A is an m x n matrix with m pivot columns, then the linear transformation x maps to Ax is a one-to-one mapping.
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2-b) If AB = C and C has 2 columns, then A has 2 columns.
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2-d) If BC = BD, then C = D.
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2-e) If AC = 0, then either A = 0 or C = 0.
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2-f) If A and B are n x n, then (A+B)*(A-B) = A^2 - B^2.
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2-j) Every square matrix is a product of elementary matrices.
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2-l) If AB = I, then A is invertible
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2-m) If A and B are square and invertible, then AB is invertible, and (AB)^-1 = A^-1*B^-1
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2-o) If A is an invertible and if r does not equal 0, then (rA)^-1 = rA^-1
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1-I) if matrices A and B are row equivalent, they have the same reduced echelon form.
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1-c) if a system of linear equations has two different solutions, it must have infinitely many solutions.
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1-e) if an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax=b and Cx=d have exactly the same solutions sets.
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1-f) if a system Ax=b has more than one solution, then so does the system Ax=0.
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1-k) IF A is an m x n matrix and the equation Ax=b is consistent for every b in Rm, then A has m pivot columns.
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1-m) if an n x n matrix A has n pivot positions, then the reduced echelon for of A is the n x n identity matrix.
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1-n) If 3 x 3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations.
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1-o) If A is an m x n matrix, if the equation Ax = b has at least two different solutions, and if the equation Ax = c is consistent, then the equation Ax = c has many solutions.
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1-p) If A and B are row equivalent m x n matrices and if the columns of A span Rm, then so do the columns of B.
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1-r) If {u,v,w} is linearly independent, then u,v,and w are not in R2.
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1-v) if w is a linear combination of u and v in Rn, then u is a linear combination of v and w.
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1-w) Suppose that v1, v2, v3 are in R5, v2 is not a multiple of v1, and v3 is not a linear combination of v1 and v2. Then {v1, v2, v3} is linearly independent.
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1-y) If A is a 6x5 matrix, the linear transformation x maps to Ax cannot map R5 to R6
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2-a) If A and B are m x n, then both A*B^T and A^T*B are defined.
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2-c) Left-multiplying a matrix B by a diagonal matrix A, with nonzero entries on the diagonal, scales the rows of B.
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2-g) An elementary n x n matrix has either n or n + 1 nonzero entries.
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2-h) The transpose of an elementary matrix is an elementary matrix.
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2-i) An elementary matrix must be square.
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2-k) If A is a 3 x 3 matrix with three pivot positions, there exist elementary matrices E1, ... , Ep such that Ep ... E1*A = I
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2-n) If AB = BA and if A is invertible, then A^-1*B = B*A^-1
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2-p) If A is a 3 x 3 matrix and the equation Ax = [1; 0; 0] has a unique solution, then A is invertible. A:
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