Reduced echelon form

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Reduced echelon form Reduced echelon form Reduced echelon form
Matrix equations Matrix equations Matrix equations
Null space Null space Null space
Range Range Range
Determinant Determinant Determinant
Invertibility Invertibility Invertibility
Similar matrices Similar matrices Similar matrices
Eigenvalues Eigenvalues Eigenvalues
Eigenvectors Eigenvectors Eigenvectors
Diagonabilty Diagonabilty Diagonabilty
Power Power Power
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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
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Matrix equations
Because the reduced echelon form of A is the
identity matrix
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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
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The determinant of A (1)(7) (4)(-2) 15
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Because the determinant of A is NOT ZERO, A is
invertible (nonsingular)
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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)
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The eigenvalues for A are 3 and 5
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A square root of A
A10
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The reduced echelon form of B
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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.
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Because the determinant is 0, B has no inverse.
ie. B is singular
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If P is a 4x4 nonsingular matrix, then B is
similar toany matrix of the form P-1 BP
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The eigenvalues are 0 and 2.
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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
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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.
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The reduced echelon form of C is
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A basis for the null space is
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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
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The determinant of C is 0. Therefore C has no
inverse.
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For any nonsingular 3x3 matrix P, C is similar
to P-1 CP
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The eigenvalues are 1, -1, and 0
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The columns of P are eigenvectors and the
diagonal elements of D are eigenvalues.
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