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Conversion of Bases
Coordinates of w.r.t. basis are written as .
Conversion from basis to basis takes on following notation:
For (standard basis), , and
Transformation matrix takes following values:
Row Space and Column Space
Span of row vectors or column vectors of matrix.
Theorem 3.6.1: Two row equivalent matrices have the same row space.
Rank of a Matrix
- the rank of A is 2
- the vectors and are linearly independent
- they form a basis for the row span of
Solving Linear System of Equations
Theorem 3.6.2: The system has a solution iff is contained in the column space of .
Let be a matrix.
- The linear system is consistent for every iff the column vectors of span .
- The system has at moste one solution for every iff the column vectors of are linearly independent.