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Definition of spanning set:
Let be vectors in the vector space . The set of all linear combination of the vectors is a subspace ( say ) of . The subspace is called the space spanned by the vectors The set is called a spanning set of .
Definition of linear independence:
Suppose that is a vector space. The set of vectors from is linearly dependent if there is a relation oflinear dependence on that is not trivial. In the case where the only relation of linear dependence on is the trivial one, then is a linearly independent set of vectors.
Definition of basis of vector space:
1. It spans the space.
2. Its vectors are independent.
3. The number of vectors in the basis is equal to the dimension of the space.
True or false? Given reason
1. A set of 5 Vectors in R5 must be a basis for R5
2. A set of 6 Vectors in R5 cannot be a basis for R5
3. A set of 7 vectors in R5 must be a spanning set for R5
4. A set of 6 polynomials in R5 must be a basis for R5
5. A set of 6 polynomials in R5 may be a basis for R5
Anyone help me to explain these question.. I want to know the reason cz it's confused me a lot and i always get mess about these kind of question