9 Compute lulg and lulg SolutionAns A basis B of a vector sp

Solution

Ans-

A basis B of a vector space V over a field F is a linearly independent subset of V that spans V.

In more detail, suppose that B = { v₁, …, v_n } is a finite subset of a vector space V over a field F (such as thereal or complex numbers R or C). Then B is a basis if it satisfies the following conditions:

for all a₁, …, a_n F, if a₁v₁ + … + a_nv_n = 0, then necessarily a₁ = … = a_n = 0; and

for every x in V it is possible to choose a₁, …, a_n F such that x = a₁v₁ + … + a_nv_n.

The numbers a_i are called the coordinates of the vector x with respect to the basis B, and by the first property they are uniquely determined.

A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite)B V is a basis, if

The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see Related notions below.

It is often convenient to list the basis vectors in a specific order, for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span V: see Ordered bases and coordinates below.

Properties[edit]

Again, B denotes a subset of a vector space V. Then, B is a basis if and only if any of the following equivalent conditions are met:

Every vector space has a basis. The proof of this requires the axiom of choice. All bases of a vector space have the same cardinality (number of elements), called the dimensionof the vector space. This result is known as the dimension theorem, and requires the ultrafilter lemma, a strictly weaker form of the axiom of choice.

Also many vector sets can be attributed a standard basis which comprises both spanning and linearly independent vectors.

Standard bases for example:

In Rⁿ, {e₁, ..., e_n}, where e_i is the ith column of the identity matrix.

In P₂, where P₂ is the set of all polynomials of degree at most 2, {1, x, x²} is the standard basis.

In M₂₂, {M_1,1, M_1,2, M_2,1, M_2,2}, where M₂₂ is the set of all 2×2 matrices. and M_m,n is the 2×2 matrix with a 1 in the m,n position and zeros everywhere else.