Statement:- The necessary & sufficient conditions for a non-empty subset W of a vector spaceV(F) to be a subspace of V are (...
Theorem-Vector Space
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Showing posts with label Linear Algebra. Show all posts
Showing posts with label Linear Algebra. Show all posts
Monday, 1 July 2013
Theorem-Vector Space
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Statement- The necessary and sufficient condition for a non-empty subset W of vector space V is that a,bєF, α,βєW ⇨ aα+bβєW, ∀ a,...
Vector Space
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Let F be an orbitrary field then a non-empty set V is called a vector space over the field F written as V(F) if following axioms...
Union Of Two Subspace
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Statement:- The union of two subspace of a vector space is a subspace iff one each contained. Proof:- Let V be a vector s...
Range,Rank,Nullity Of A Linear Transformation
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Range of a Linear Transformation:- Let U and V be two vector spaces over the field F.Let T be a linear transformation from U into V...
One-One & Onto Linear Transformation
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One-one transformation:- Let T be a transformation from a vector space U into V then T is said to be one-one transformation if α ...
Null Space of a Linear Transformation
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Let U and V be two vector spaces over the same field F and let T be a linear transformation from U into V. The null space of T is t...
Linear Transformation
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Linear Transformation:- Let U & V be two vector spaces over the same field F.A linear transformation from U into V is a functio...
Linear Sum Of Two Subspaces
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Let V be a vector space over the field F the linear sum of two subspaces W 1 and W 2 of V written as (W 1 +W 2 ) and is defined as W ...
Linear combination & Span
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Linear Combination Of Vectors: - Let V be a vector space over the field F,then a vector αєV is said to be a linear combination of v...
Invarience Theorem
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Statement:- Any two bases of a finite dimensional vector space have same number of elements. OR The number of elements in a basi...
Intersection Of A Vector Space
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Statement:- The intersection of any two sub-spaces of a vector space is also a subspace of the same vector space. Proof:- L...
Inner Product Space
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Let V be the vector space over the field F where F is either field of real numbers or field of complex number. An inner produc...
Statement:-There exists a basis for each finite dimensional vector space.
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Proof:- Let V be vector space & let S={ α 1 , α 2 ,…………….., α m } be a finite subset of V such that L(S)=V. If S is linearly inde...
Theorem-Direct Sum
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Statement:- The necessary and sufficient condition for a vector space V over the field F to be direct sum of its two subspaces W 1 an...
Direct Sum & Disjoint Subspace
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Direct sum of a vector subspace:- Let V be a vector space over the field F then vector space V is said to be direct sum of its su...
Theorem-Dimension Of A Vector Space
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Theorem:- If W is a subspace of an n-dimensional vector space over the field F then dimW≤dimV
Basis Dimension & Cosets Of A Vector Space
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Basis of a vector space:- A non-empty subset S of a vector space V(F) is said to be its basis if 1-S is linearly independent. ...
Annihilators
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If V is a vector space over the field F and S is a subset of V,the annihilator of S is the set S o of all linear functional f on V suc...
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