12. a. If there is an n x n matrix D such that AD = 1, then there is also an n x n matrix C such that CA= 1.
b. If the columns of A are linearly independent, then the columns of A span Rn.
c. If the equation Ax = b has at least one solution for each bin Rn, then the solution is unique for each b.
d. If the linear transformation (x) -> Ax maps Rn into Rn, then A has n pivot positions.
e. If there is a b in Rn such that the equation Ax = b is inconsistent, then the transformation x -> Ax is not one- to-one.
Let A be an nx n matrix. Select all of the following that are equivalent to the statement: A is invertible. The homogeneous equation Ax-0 has a nontrivial solution. The echelon form of A has a pivot in every row and every column. The columns of A are linearly dependent For any vector b in R", Ax-b has a unique solution. The linear transformation x Ax is 1-1 and onto. A is nonsingular.
8. Let A be a 5 x 4 matrix such that its reduced row echelon form has 4 pivot positions (leading entries). Which of the following statements is TRUE? a) The linear transformation T : R4 → R5 defined by T(X) = AX is onto. b) AX = 0 has a unique solution. c) Columns of A are linearly dependent. d) AX b is consistent for every vector b in R
a.) if A is an m*n matrix, such that Ax=0 for every vector x in R^n, then A is the m * n Zero matrix b.) The row echelon form of an invertible 3 * 3 matrix is invertible c.) If A is an m*n matrix and the equation Ax=0 has only the trivial solution, then the columns of A are linearly independent. d.) If T is the linear transformation whose standard matrix is an m*n matrix A and the...
True/False: Give a brief justification for your answer a) If an m x n matrix A has a pivot position in each row, then the equation Ax=b has a unique solution for each b in R^m. b) If {u,v,w} is linearly independent, then u, v, w are not in R^2. c) If A is a 5 x 4 matrix, then the linear transformtion x -> Ax is not onto.
IT a) If one row in an echelon form for an augmented matrix is [o 0 5 o 0 b) A vector bis a linear combination of the columns of a matrix A if and only if the c) The solution set of Ai-b is the set of all vectors of the formu +vh d) The columns of a matrix A are linearly independent if the equation A 0has If A and Bare invertible nxn matrices then A- B-'is the...
Write each statement as True or False (a) If an (nx n) matrix A is not invertible then the linear system Ax-O hns infinitely many b) If the number of equations in a linear system exceeds the number of unknowns then the system 10p solutions must be inconsistent ) If each equation in a consistent system is multiplied through by a constant c then all solutions to the new system can be obtained by multiplying the solutions to the original...
Plese help me!!!(Conditioning of Problems and Stability of Algorithms) IA is an m x n matrix, and x is an n x 1 vector, then the linear transformation У-Ax maps Rn to Rm, so the linear transformation should have a condition number, condAar (x). Assume that ||l is a subordinate norm. a. Show that we can define condAx (x) = 11All 11제/IAxl for every x 0. IA is an m x n matrix, and x is an n x 1...
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer. 0 3 - 4 0 2 -4 -9 4 Choose the correct answer below. O A. The matrix is not invertible. If the given matrix is A, the equation Ax=b has at least one solution for each b in R3. OB. The matrix is invertible. The given matrix has 3 pivot positions. C. The matrix is invertible. The columns of the given matrix span...
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer. - 3 30 20 6 -40 9 Choose the correct answer below. O A. The matrix is invertible. The given matrix has 2 pivot positions. O B. The matrix is not invertible. If the given matrix is A, the columns of A do not form a linearly independent set. OC. The matrix is not invertible. If the given matrix is A, the equation Ax...
Currently workable: Suppose and m x n matrix A has n pivot columns. Prove why, for each b in R the equation Ax = b has at most one solution. + Drag and drop your files or click to browse...