Download - Linear Algebra and Real Analysis 1
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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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4 x 1
4 2 3
x 3 x
x : epsilon cannot be assigned as they form a closed loop with other assigned cells..
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
..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... 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
..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 boundedso, if f is not bounded f cannot be R integrable.
However, bounded functions need not be R integrable. (ex. Dirichlets function).