Math 220 - April 9, 1999

Bases and Isomorphisms

• Proposition. Let V be a given vector space and e = { e_{1}, ..., e_{n} } a given sequence of vectors in V. Let phi_{e} be the linear map from R^{n} to V that is defined by the formula phi_{e}(x_{1}, ..., x_{n}) = x_{1} e_{1} + ... x_{n} e_{n}. Then

  1. The linear map phi_{e} is injective if and only if the sequence e is linearly independent.

  2. The linear map phi_{e} is surjective if and only if the sequence e spans V.

  3. The linear map phi_{e} is invertible if and only if the sequence e is a basis of V.

• Definitions. An isomorphism is an invertible linear map between vector spaces. If there exists an isomorphism between two vector spaces, the vector spaces are said to be isomorphic. An invertible linear map from a vector space to itself is called an automorphism. (An automorphism is considered to be an isomorphism.)

• Theorem. Let V be a given vector space. The following items of data are transparently equivalent:

  1. An isomorphism from R^{n} to V.

  2. A basis of V.

  3. A coordinate system in V whose origin is the zero element of V. (But what is the definition of ``coordinate system''?)

• Corollary. A vector space is finite-dimensional if and only if it is isomorphic to R^{n} for some (unique) n.

• Corollary. A basis of R^{n} is the same thing as the sequence of columns of an invertible matrix.

• Corollary. Let V be a given vector space and e_{1}, ..., e_{n} a given basis of V. (Then dim V = n. ) The following items of data are transparently equivalent:

  1. An n \times n matrix.

  2. A linear map from V to V.

  Moreover, if the matrix has rank r, then one has:

  1. The dimension of the image of the linear map is r.

  2. The linear map is an automorphism if and only if r = n.

  One says that the matrix is the matrix of the linear map with respect to the basis e_{1}, ..., e_{n}. The matrix is characterized by the statement that the j^{th} column of the matrix is the column of coefficients, relative to the basis, of the point that results from applying the map to the j^{th} member of the basis.

Assignment for Monday, April 12

1. Let e_{1} = (1, 0) and e_{2} = (0, 1) be the standard basis of the Cartesian plane. Find the matrix of the rotation about the origin through the angle theta relative to this basis.

2. Find the matrix with respect to the basis e in the previous exercise of the reflection in the line through the origin that has angle of elevation theta/2 (counterclockwise from the positive direction along the first coordinate axis).

3. When g is the basis of the Cartesian plane with g_{1} = (2, 2) and g_{2} = (-2, 2) what is the matrix of the rotation about the origin through the angle pi/3 relative to g ?

4. When h is the basis of the Cartesian plane with h_{1} = (a, b) and h_{2} = (c, d), what is the matrix of the rotation about the origin through the angle pi/3 relative to h ? (Assume that a d - b c <> 0. )