Vector Coordinate System

• Let be an ordered basis for a finite dimensional Vector Space . For we define the coordinate vector of relative to denoted by

  where .

  • (Friedberg e2.2.7) Let be an -dimensional vector space with an ordered basis . Define by . is linear.
  • The standard representation of is denoted and is defined by
    • For any finite-dimensional vector space and ordered basis , is an isomorphism.

• Let and be finite-dimensional vector spaces with ordered basis and .

  Let be linear. Then, we have as scalars such that

  The Matrix defined by is the matrix that represents in the ordered bases and . We denote this .

  If and , then we write

Change of Coordinates §

• (Friedberg 2.23) Let and be two ordered bases for a finite dimensional vector space . Let Then

  • is invertible
  • ,

• The matrix is called the change of coordinate matrix. It changes ’ coordinates into coordinates

• (Friedberg 2.24) Let be a linear transformation of the finite-dimensional vector space and let and be ordered bases for . Let be the change of coordinate matrix which changes -coordinates into coordinates. Then

  • (Friedberg e2.5.7) Let be a linear transformation from a finite-dimensional vector space to a finite dimensional vector space . Let be ordered bases for and be ordered bases for . Then

    Where is the change of coordinates matrix from -coordinates to -coordinates

    And is the change of coordinates matrix from -coordinates into coordinates

• Let and be elements of . We say that is similar to if there exists an invertible matrix such that

  Denote

  • (Friedberg e2.5.9) If then

  • (Friedberg 5.1) Let and be any ordered basis for . Then

    Where is the matrix whose columns are the entries in (i.e., it is the change of coordinate matrix)

    Conversely, (Friedberg 5.2) For , where and is the ordered basis for . If is defined such that , then there exists a basis for such that

  • (Friedberg e5.1.19) Let such that . There exists an -dimensional vector space , a linear operator of and bases of such that and .

• (Friedberg e2.5.12) Let be a finite-dimensional vector space over a field and be an ordered basis for . Let be an invertible matrix with entries from . Define

  And set . is a basis for and is the change of coordinate matrix changing -coordinates into coordinates

Links §

• Friedberg, Insel and Spence
• Linear Transformation Matrix Isomorphism