Day 28: Linear Algebra

• The book has been updated (unbeknownst to me! Grrrr....).
• Hence problem numbers do not correspond to those I'm using.
• Until I hear back from the author about the extent of the changes, I'll continue to use the original copy that we used at the start of the semester.
• There are links to the original and the answers on our course webpage.

"...a linear map is determined by its action on a basis."

In this section we re-express homomorphisms as matrices.

• Summary: A homomorphism is determined by its action on a basis. That is, if is a basis of a vector space V and are (perhaps not distinct) elements of a vector space W, then there exists a homomorphism from V to W sending to , and that homomorphism is unique.

  Let's take a good look at example 1.1 from this section, p. 193.

  At the end of this example, bottom of p. 194, the author defines (essentially creates a) matrix multiplication. This definition is crucial for all that follows.

• Definitions:
  1. matrix representation of homomorphism h with respect to B, D (bases of domain and codomain, respectively:
     

  2. matrix multiplication:

     

     Observe the essential alignment of dimensions:

     • the number of rows in the vector multiplied by the matrix must conform to the number of columns of the matrix.
     • The number of rows of the result will conform to the number of rows of the matrix.
     "A good way to view a matrix-vector product is as the dot products of the rows of the matrix with the column vector."
     

     "Matrix-vector product can also be viewed column-by-column."

     

• Theorems:
  1. Theorem 1.4:
     

• Examples:
  1. Really important example: 1.8, p. 198
  2. 1.11, p. 200 -- let's think of the multiplication in as many different ways as possible...
  3. 1.13 -- flashback to linear systems, to column spaces, and now to thinking of matrix multiplication as a linear map taking vectors in one space to another space.

     Remember: all finite dimensional vector spaces are essentially equivalent!

• Summary: "The prior subsection shows that the action of a linear map h is described by a matrix H, with respect to appropriate bases.... In this subsection, we will show the converse, that each matrix represents a linear map."

  "....In practice, when we are working with a matrix but no spaces or bases have been specified, we will often take the domain and codomain to be and and use the standard bases."

• Definitions:

• Theorems:
  1. Theorem 2.1: Any matrix represents a homomorphism between vector spaces of appropriate dimensions, with respect to any pair of bases.

  2. Theorem 2.3: The rank of a matrix equals the rank of any map that it represents.

  3. Corollary 2.5: Let h be a linear map represented by a matrix H. Then h is onto if and only if the rank of H equals the number of its rows, and h is one-to-one if and only if the rank of H equals the number of its columns.

  4. Corollary 2.6: A square matrix represents nonsingular maps if and only if it is a nonsingular matrix. Thus, a matrix represents an isomorphism if and only if it is square and nonsingular.

• Examples:
  1. Let's look at a better example, perhaps, to motivate Example 2.2, p. 204.
  2. 2.10a, p. 207
  3. 2.11 -- notice that our author relates the mapping back to the standard basis. It's our preferred way of thinking about
  4. 2.12