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Reduced echelon form

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Reduced echelon form

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  1. Reduced echelon form Because the reduced echelon form of A is the identity matrix,we know that the columns of A are a basis for R2 Return to outline

  2. Matrix equations Because the reduced echelon form of A is the identity matrix: Return to outline

  3. Return to outline

  4. Every vector in the range of A is of the form: Is a linear combination of the columns of A. The columns of A span R2 = the range of A Return to outline

  5. The determinant of A = (1)(7) – (4)(-2) = 15 Return to outline

  6. Because the determinant of A is NOT ZERO, A is invertible (nonsingular) Return to outline

  7. If A is the matrix for T relative to the standard basis,what is the matrix for T relative to the basis: Q is similar to A. Q is the matrix for T relative to the  basis, (columns of P) Return to outline

  8. The eigenvalues for A are 3 and 5 Return to outline

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  11. A square root of A = A10 = Return to outline

  12. The reduced echelon form of B = Return to outline

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  14. Return to outline

  15. The range of B is spanned by its columns. Because its null spacehas dimension 2 , we know that its range has dimension 2.(dim domain = dim range + dim null sp).Any two independent columns can serve as a basis for the range. Return to outline

  16. Because the determinant is 0, B has no inverse. ie. B is singular Return to outline

  17. If P is a 4x4 nonsingular matrix, then B is similar toany matrix of the form P-1 BP Return to outline

  18. The eigenvalues are 0 and 2. Return to outline

  19. The null space of (2I –B)=The eigenspace belonging to 2 The null space of (0I –B)= the null space of B.The eigenspace belonging to 0= the null space of the matrix Return to outline

  20. There are not enough independent eigenvectors to make a basis for R4 . The characteristic polynomial root 0 is repeated three times, but the eigenspace belonging to 0 is two dimensional. B is NOT similar to a diagonal matrix. Return to outline

  21. The reduced echelon form of C is Return to outline

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  23. A basis for the null space is: Return to outline

  24. The columns of the matrix span the range. The dimension of the null space is 1. Therefore the dimension of the range is 2. Choose 2 independent columns of C to form a basis for the range Return to outline

  25. The determinant of C is 0. Therefore C has no inverse. Return to outline

  26. For any nonsingular 3x3 matrix P, C is similar to P-1 CP Return to outline

  27. The eigenvalues are: 1, -1, and 0 Return to outline

  28. Its null space = Its null space = Its null space = Return to outline

  29. The columns of P are eigenvectors and the diagonal elements of D are eigenvalues. Return to outline

  30. Return to outline