Reduced echelon form

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

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

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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

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

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

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

The eigenvalues for A are 3 and 5 Return to outline

A square root of A = A10 = Return to outline

The reduced echelon form of B = Return to outline

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

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

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

The eigenvalues are 0 and 2. Return to outline

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

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

The reduced echelon form of C is Return to outline

A basis for the null space is: Return to outline

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

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

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

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

Its null space = Its null space = Its null space = Return to outline

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

Reduced echelon form

Reduced echelon form

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