The following table gives the approximate height in meters a

The following table gives the approximate height (in meters) at quarter-meter Intervals of a mound of gravel. Estimate the volume V of the mound by computing the average of the two Riemann sums S_4, 3 with lower-left and upper-right vertices of the subrectangles as sample points.

Solution

Ans-

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 dimension of 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.

Change of basis

Main article: Change of basis

Given a vector space V over a field F and suppose that {v₁, ..., v_n} and {₁, ..., _n} are two bases for V. By definition, if is a vector in V then = x₁₁ + ... + x_nn for a unique choice of scalars x₁, ..., x_n in F called the coordinates of relative to the ordered basis {₁, ..., _n}. The vector x = (x₁, ..., x_n) in Fⁿ is called the coordinate tuple of (relative to this basis). The unique linear map : Fⁿ V with (v_j) = _j for j = 1, ..., n is called the coordinate isomorphism for V and the basis {₁, ..., _n}. Thus (x) = if and only if = x₁₁ + ... + x_nn.

A set of vectors can be represented by a matrix of which each column consists of the components of the corresponding vector of the set. As a basis is a set of vectors, a basis can be given by a matrix of this kind. The change of basis of any object of the space is related to this matrix. For example, coordinate tuples change with its inverse.

Examples