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Suppose V is finite dimensional vector space over R .if W1 is subspace of V then prove that W1 not equal to V ,then span (V\W1)=V?
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Suppose V is finite dimensional vector space over R .if W1 is subspace...
Proof:
Given:
V is a finite-dimensional vector space over R.
W1 is a subspace of V.
To Prove:
If W1 is not equal to V, then span(V\W1) = V.
Proof:
1. W1 is a Proper Subspace:
Since W1 is not equal to V, it implies that W1 is a proper subspace of V.
2. Dimension of V:
Since V is a finite-dimensional vector space, let dim(V) = n.
3. Basis of W1:
Let {v1, v2, ..., vk} be a basis for W1, where k < />
4. Extension to Basis of V:
Since W1 is a proper subspace, we can extend the basis {v1, v2, ..., vk} to a basis of V, say {v1, v2, ..., vk, u1, u2, ..., um}, where m = n - k.
5. Span of V\W1:
The set {u1, u2, ..., um} forms a basis for the span of V\W1, as any vector in V not in W1 can be expressed as a linear combination of {u1, u2, ..., um}.
6. Conclusion:
Since {u1, u2, ..., um} is a basis for the span of V\W1, and it spans all vectors in V not in W1, we can conclude that span(V\W1) = V.
Therefore, if W1 is not equal to V, then span(V\W1) = V.
Given:
V is a finite-dimensional vector space over R.
W1 is a subspace of V.
To Prove:
If W1 is not equal to V, then span(V\W1) = V.
Proof:
1. W1 is a Proper Subspace:
Since W1 is not equal to V, it implies that W1 is a proper subspace of V.
2. Dimension of V:
Since V is a finite-dimensional vector space, let dim(V) = n.
3. Basis of W1:
Let {v1, v2, ..., vk} be a basis for W1, where k < />
4. Extension to Basis of V:
Since W1 is a proper subspace, we can extend the basis {v1, v2, ..., vk} to a basis of V, say {v1, v2, ..., vk, u1, u2, ..., um}, where m = n - k.
5. Span of V\W1:
The set {u1, u2, ..., um} forms a basis for the span of V\W1, as any vector in V not in W1 can be expressed as a linear combination of {u1, u2, ..., um}.
6. Conclusion:
Since {u1, u2, ..., um} is a basis for the span of V\W1, and it spans all vectors in V not in W1, we can conclude that span(V\W1) = V.
Therefore, if W1 is not equal to V, then span(V\W1) = V.
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Suppose V is finite dimensional vector space over R .if W1 is subspace of V then prove that W1 not equal to V ,then span (V\W1)=V?
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Suppose V is finite dimensional vector space over R .if W1 is subspace of V then prove that W1 not equal to V ,then span (V\W1)=V? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Suppose V is finite dimensional vector space over R .if W1 is subspace of V then prove that W1 not equal to V ,then span (V\W1)=V? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose V is finite dimensional vector space over R .if W1 is subspace of V then prove that W1 not equal to V ,then span (V\W1)=V?.
Suppose V is finite dimensional vector space over R .if W1 is subspace of V then prove that W1 not equal to V ,then span (V\W1)=V? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Suppose V is finite dimensional vector space over R .if W1 is subspace of V then prove that W1 not equal to V ,then span (V\W1)=V? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose V is finite dimensional vector space over R .if W1 is subspace of V then prove that W1 not equal to V ,then span (V\W1)=V?.
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