Brief Notes/code
Monday September 13
Solving AX=B. One can find solutions by row reducing the augmented matrix A:B. Consistent, inconsistent systems. If x_p is one solution to AX=B , then any solution to AX=B is x_h+x_p where x_h is some solution to AX=0.
MATLAB: apply rref to the augmented matrix A B. Or use A\B. (But this may get into some numerical analysis issues and may give inaccurate results. Especially if Sol(A) is other than just zero.
>> A=[1 2 3; 2 3 4; 5 6 7]
>> X=1/A
>> B=[1 2 3]'
>> rref([A B])
ans =

     1     0    -1     0
     0     1     2     0
     0     0     0     1
(This shows an inconsistent system because of the last row)
>> X=A\B
Warning: Matrix is close to singular or badly scaled.
         Results may be inaccurate. RCOND = 7.930164e-18.
X =
   1.0e+15 *

    1.1259
   -2.2518
    1.1259

>> det(A)
ans = 0
(this tells me Sol(A) is more than 0)

>> pinv(A)*B (You can use this to find one particular solution (if there is a solution) and then any solution is this plus what you get from null(A))

ans =

   -0.0000
    0.1538
    0.3077
(The above is a bogus answer. Things went awry because the system is inconsistent. Moral: be a good engineer and check your work.)

>>
Friday September 10
A an m by n matrix. We find Sol(A), the  subspace of Rⁿ consisting of the solutions to AX=0. We already saw the previous Friday how to find this from A_Rand will do that again here. Sol(A) has dimension n-rank(A). This is because n = # leading variables in A_R+ # non leading variables in A_R .The first summand is the rank of A and also the number of dependent variables; the second is the number of independent variables. The MATLAB command to find Sol(A) is null(A),
Wednesday September 8
A an m by n matrix. Row(A) is the subspace of  Rⁿ spanned by the rows of A. It has dimension at most m and is equal to Row(A_R). Row(A_R) has basis its nonzero rows. Thus to find Row(A) using MATLAB type in rref(A) and then your basis is the nonzero rows of rref(A). col(A) is the subspace of  R^m spanned by the columns of A . Its dimension is the same as that of Row(A) and is called the Rank of A. Thus Rank(A) is no bigger than the smaller of m and n. It is not true that Col(A)=Col(A_R). You can find the row reduced form of A' (the transpose of A) and the  rows of this give a basis for col(A). So to find col(A) type in rref(A') and your basis are the rows of the resulting matrix. Often one find row(A) by row reducing and then, since you know what the rank of A is, you kknow what the dimension of col(A) is. And you can pick that many linearly independent columns vectors form the original A.
Friday September 3
Row reduced escelon form. rref(A).   Solving AX=0 when in rref.
Wednesday Sept 1
Transpose, Adjacency matrix of a graph. Walks of length k between two vertices. Elementary row operations. These do not effect the solution to the associated system of homogeneous equations. Elementary matrices.
Monday Aug 30
Went over homework for 6.5. Discussed matrices and operations on them. Examples of matrix multiplication. Matrix multiplication is not commutative. Representing a linear system of equations using matrices .
Wednesday Aug 25, Friday Aug 27
6.5 "building subspaces of Rⁿ :  Span of a set of vectors is all linear combinations of those vectors. Spanning set for a subspace. Linear dependence, linear independence. A minimal spanning set is called a basis.It's elements must be linearly independent. The number of elements in a basis for a subspace is the same for any basis for that subspace and is called the dimension of the subpace. Special cases where a set of vectors is linearly independent. Examples. Quiz on Friday.
Monday Aug 23
Syllabus, topics covered in the course, reason they are important,examples
6.4: vectors, operations. This makes Rⁿ  a vector space. Metric properties. Standard basis. Subspace: definition (a subset which is a vector space),examples, nonexamples, subspaces of R²,R³
Matlab: sphere(n)