Day 20: Linear Algebra

Comment on #34 -- what an odd set of operations!

• What do we need to do in this problem?
  1. Find a spanning set.
  2. Show that it's independent.

• What is the zero vector?

• Definitions:
  1. row space: The row space of a matrix is the span of the set of its rows. The row rank is the dimension of the row space, the number of linearly independent rows.
  2. column space: The column space of a matrix is the span of the set of its columns. The column rank is the dimension of the column space, the number of linearly independent columns.
  3. transpose: The transpose of a matrix is the result of interchanging the rows and columns of that matrix. That is, column j of the matrix A is row j of A^trans , and vice versa.
  4. rank: The rank of a matrix is its row rank or column rank (they're the same!).

• Theorems:
  1. Lemma: row-equivalent matrices have the same row space, and hence also, the same row rank.
  2. Lemma: The nonzero rows of an echelon form matrix make up a linearly independent set.
  3. Lemma: Row operations do not change the column rank.
  4. Theorem: The row rank and column rank of a matrix are equal.
  5. Theorem: For linear systems with n unknowns and with matrix of coefficients A, the statements
     1. the rank of A is r
     2. the space of solutions of the associated homogeneous system has dimension n-r
     are equivalent.
  6. Corollary (important! It's kind of buried in the text on p. 126): For the matrix A_nxn, the statements
     1. the rank of A is n
     2. A is non-singular
     3. the rows of A form a linearly independent set
     4. the columns of A form a linearly independent set
     5. any linear system whose matrix of coefficients is A has one and only one solution
     are equivalent.

• Examples:
  1. 3.16ad, p. 127
  2. 3.17a
  3. • Notice that row-reduction doesn't necessarily result in a basis for the column space. In this case, the column space is fundamentally changed by the row-reduction.

       But transposing the matrix followed by row-reduction will give a basis for the column space:
     • Notice that Octave doesn't care if I use commas between row elements or not.

  4. 3.26
  5. 3.33
  6. 3.36

• Definitions:
  1. isomorphism: An isomorphism between two vector spaces V and W is a map that
     1. is one-to-one and onto from one set of vectors to the other; and
     2. preserves structure (the two operations): if and if where it is important to realize that the two operations of addition and multiplication by a scalar are generally different in the two spaces.
     We write , read "V is isomorphic to W", when such a map exists.

  2. automorphism: an isomorphism between a space and itself.

• Theorems:
  1. 1.8 Lemma: An isomorphism maps the zero vector of V to the zero vector of W.

  2. 1.9 Lemma: For any map betweeen vector spaces the following are equivalent:
     1. f preserves structure (the two operations): and
     2. f preserves linear combinations:
     3. f preserves linear combinations of any finite number of vectors:

  These two lemmas help us in two ways:
  1. The first gives us a quick check on a potential isomorphism: does it map one zero vector to the other zero vector? If not, it's not an isomorphism.
  2. The second gives us a check for the preservation of operations: just check that linear combinations are preserved (the second part above).

• Examples:
  1. Is it obvious that the space of points in the Cartesian plane with the usual operations is isomorphic to the space of points with the usual operations?

     Give an example mapping. That is, explained how vectors are carried into vectors, preserving operations.

     This is an example of an automorphism.

  2. 1.11, p. 164
  3. 1.13ad
  4. 1.27
  5. 1.21