Linear Transformation Matrix Isomorphism

• Every Linear Transformation is associated with a Matrix (see more here). Similarly, every matrix is associated with a linear transformation. This mapping is isomorphic. Results from matrices can be generalized to linear transformation and vice versa.

• (Friedberg 2.8) Let and be finite dimensional vector spaces with ordered bases and and be linear. Then

• We have an isomorphism between the set of matrices and the set of linear transformations.

• Composition of linear transformations is analogous to Matrix Multiplication.

  • (Friedberg 2.9) Let be vector spaces and , be linear. Then is linear.

  • (Friedberg 2.10) Let be a vector space and . Then

    • .

  • (Friedberg 2.11) Let be finite-dimensional vector spaces with ordered bases respectively. Let and be linear. Then

    An immediate corollary of this if all bases are the same

  • (Friedberg 2.12)

• (Friedberg 2.15) Let and be finite dimensional vector spaces having ordered bases and respectively and let be linear. Then for each

  • Another way to say this is

• Let . We denote , defined by for each . We call a left multiplication transformation

• (Friedberg 2.16) Let . Then the left-multiplication transformation is linear. Furthermore, if is any other matrix with entries from and and are the standard ordered bases for and respectively, then

  • If is linear, then there exists a unique matrix such that . In fact,
  • If then
  • If , then

• (Friedberg 3.7) Let and be linear transformations on finite dimensional vector spaces and and let be matrices such that is defined.

• (Friedberg e3.2.14) Let

  • If is finite dimensional, then
  • For any

Invertibility §

• Let and be vector spaces and let be linear. has an inverse if and . The inverse is unique and we denote it as . We say that is invertible if has an inverse

  The following hold true for invertible functions

  • . is also invertible.

• (Friedberg 2.18) Let be vector spaces and be linear and invertible. Then is also linear.

• If is a linear transformation between vector spaces of equal finite dimension, then being invertible, one-to-one, and onto are equivalent (see Friedberg 2.5)

• (Friedberg 2.19Lem) Let be finite dimensional and be linear. If is invertible, then

• (Friedberg 2.19) Let be finite ordered bases for and respectively and be linear. Then is invertible if and only if is invertible. In particular

• (Friedberg 2.19.1) Let be a finite dimensional vector space with ordered basis and be linear. Then is invertible if and only if is invertible. Also

• (Friedberg 2.19.2) Let . is invertible if and only if is invertible. Also

• (Friedberg 2.21) Let be finite dimensional vector spaces over with and . Let be ordered bases for and respectively. Then the function defined by

  for is an isomorphism.

  • (Friedberg 2.21.1) is dimensional of dimension

Links §

• Friedberg, Insel and Spence