2/4/2015
linear dependence, linear independence, basis, dimension, examples.
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2/2/2015
vector subspace, linear combination, linear span of a subset of a vector space, 
examples

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1/30/2015
Definition of a vector space (with the field of scalars being either
the field of all real numbers or all complex numbers)

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1/28/2015
Review the definitiion of determinant, properties of determinants,
computing determants using row/column operations.

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January 26, 2015
1. Special cases when the inverse of a square matrix A can be computed easily:
(a) A is 2-by-2 (ther is a simple formula)
(b) A is orthogonal (then the transpose of A is the inverse of A)
(b') A is unitary (then the transpose conjugate of A is the inverse of A)
(c) A=I+N, N^n=0, then I-N+N^2+...+(-1)^{n-1} N^{n-1} is the inverse of A

2. sign of permutations; definition of determinant as a sum of n! terms index by permutations, each term of which is a product of n entries of the nxn matrix 
multiplied by the sign of the permutation.
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1/23/2015

Inverse of square matrices: definition and how to compute it using row operations
criterion for a matrix to be invertible: procedurally (using row operations) and
conceptually (rank = rank of augmented matrix)
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1/21/2015

Rank of matrices, row rank = column rank, 
meaning of rank for system of linear equations

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1/16/2015
Systems of linear equations, represented using matrix multliplication,
Gaussian/row operations illustrated with an example, how to write down the 
independent solutions once one gets to the row-reduced echelon form

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1/14/2015
definition of matrix multiplication, 
structural way of looking at the matrix multiplication: by columns and by rows