VS10
Consider the set of (column) vectors defined ty X = {x ∈ R3 | x1 + x2 +x3 = 0, where xT = [x1 , x2 , x3 ]T}, which of the following is TRUE? [CS: GATE-2007]

(a) {[1, –1, 0]T, [1, 0, –1]T} is a basis for the subspace X.
(b) {[1, –1, 0]T, [1, 0, –1]T} is linearly independent set, but it does not span X and therefore is not a basis of X
(c) X is not a subspace for R3
(d) None of the above


Vector Space V – Span , Basis and Orthonormal basis
A collection of vectors which is closed under addition and scaler multiplication is called a vector space.

Closed under addition and scaler multiplication implies that linear combination of any number of vectors in the vector space also lies in the vector space.

Span – A collection of vectors v1, v2, …vk from vector space V is said to span V if every vector in V can be written as linear combination of v1, v2, …vk. (Note that v1, v2, …vk do not have to be linearly independent).

Basis – A collection of linearly independent vectors v1, v2, …vn that spans vector space V is called basis of vector space V. Basis for a vector space is not unique but number of vectors in the basis is fixed. This fixed number of vectors in basis is called dimension of vector space dim V or dim(V).

Every vector in vector space V can be written as linear combination of basis in a unique way. The scalers in the linear combination are called coordinates of the vector with respect to the basis.

Orthonormal basis – If the vectors in the basis are orthogonal to each other and have unit magnitude then the basis is called orthonormal basis. Any basis can be converted into orthonormal basis.




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