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Equivalent conditions for Bases
Transcript of Equivalent conditions for Bases
Any two of the following are true:
Linear system conditions
is a basis for V
has exactly one solution
The linear system
has exactly one solution for any a, b, c, ...
has nonzero determinant
has an inverse
If you are working in a vector space that is not n-dimensional Euclidean, you can fill in the coordinates of your basis with respect to the standard basis as the columns of this matrix.
Definition of basis.
Definition of span and linear independence
Translation into solving a linear system.
Condition for a linear system always having one solution.
condition for a matrix to be invertible.
The single solution is:
definition of dimension and PSet 7, Problem #6