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Similar matrices have same eigenvalues but different eigenvectors
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Darboux theorem
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Krishna series page 20
are basic feasible solutions
cannot be basic feasible solution as it fails to be non-negative
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same eigenvalues (with same algebraic multiplicity) A and B need not be similar.
x2=2, x3=0, x4=0 ; x1=1 u1 =(1, 2, 0, 0)
x2=0, x3=2, x4=0 ; x1= -3 u2 =(-3, 0, 2, 0)
x2=0, x3=0, x4=1 ; x1=0 u3 =(0, 0, 0, 1)
they are linearly independent
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Thomas 855
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Usually transportation problem is a minimization problem, where the objective is to minimize the transportation cost.
When the objective is to maximize then
subtract each element of the transportation matrix by the largest element.
8
4
x
1
4
2
3
x
3
x
x : epsilon cannot be assigned as they form a closed loop with other assigned cells
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is uniformly continuous
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or
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uniform continuity need not imply differentiability
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Second method:
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is discontinuous everywhere
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Example of a function which is derivable but its derivative is not derivable
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use Dinis Theorem
fn f pointwise
each fn is continuous and f is continuous
fn is monotonic
fn is monotonically increasing
to s.t fn is bounded above
squaring both sides
fn is monotonically increasing and bounded above , hence convergent (pointwise)
to find the limit
and f(x) is continuous.
by Dinis theorem : fn converges to f uniformly
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both f and g are integrable, but
which is Dirichlets function, is not integrable
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Rank of a Skew-Symmetric Matrix is always even.
Rank of a skew-symmetric matrix is atleast 2.
the smallest minor with non-zero determinant
hence rank is atleast 2.
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i.e
composition of two linear transformations is again linear.
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(try for 2 x 2 case)
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Diagonalise RHS matrix
and
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apply log test
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second log test
replace a by x
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by cauchys nth root test
this is equivalent to
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solution 1:
2)
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non-homogenous equations
find the condition for this to be consistent.
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f is R integrable f is bounded
so, if f is not bounded f cannot be R integrable.
However, bounded functions need not be R integrable. (ex. Dirichlets function)
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