MATH 323 Lecture 14

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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 .

Theorem 3.6.3

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.