Reduced echelon form

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# Reduced echelon form - PowerPoint PPT Presentation

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

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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

Matrix equations

Because the reduced echelon form of A is the identity matrix:

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

Because the determinant of A is NOT ZERO,

A is invertible (nonsingular)

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)

A square root of A =

A10 =

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.

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

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

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.

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

The determinant of C is 0.

Therefore C has no inverse.