Web viewf x = lim n→∞ f n x = lim n→∞ nx 1+ n 2 x 2 = lim n→∞ x 1 n...

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Transcript of Web viewf x = lim n→∞ f n x = lim n→∞ nx 1+ n 2 x 2 = lim n→∞ x 1 n...

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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.

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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

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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:

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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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