Monday September 13
Solving AX=B. One can find solutions by row reducing the augmented
matrix A:B. Consistent, inconsistent systems. If xp is one
solution to AX=B , then any solution to AX=B is xh+xp
where xh 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]
>> B=[1 2 3]'
>> rref([A B])
1 2 0
0 0 1
(This shows an inconsistent system because of the last row)
Warning: Matrix is close to singular or badly scaled.
Results may be
inaccurate. RCOND = 7.930164e-18.
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))
(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
the solutions to AX=0. We already saw the previous Friday how to
find this from ARand will do that again here. Sol(A) has
dimension n-rank(A). This is because n = # leading variables in AR+ # non leading variables in AR
.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(AR). Row(AR) 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 Rm
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(AR). 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
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
Monday Aug 23
Syllabus, topics covered in the course, reason they are
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