Linear Algebra Vector Spaces 1 Give three examples of a vec

Linear Algebra - Vector Spaces

1) Give three examples of a vector space with dimension 5. Give a basis for your examples

2) Give two examples of a vector space with infinite dimension. Give a basis for your examples

Solution

Vector Spaces of Dimension 5:

Example 1:

The subspace of R⁵ consisting of matrices of the type ( a, b, c, d,e)^T is a vector space of dimension 5 with { 1, 0,0, 0, 0}^T , { 0, 1,0,0,0}^T , { 0, 0,1,0,0}^T, { 0,0,0,1,0}^T and { 0,0,0,0,1}^T as basis.

Example 2:

The subspace of R⁵ consisting of all polynomials of degree 5 is a vector space with { 1, x, x² , x³, x⁴ , x⁵} as basis.

Example 3:

A vector space V composed of 5-tuples (sequences of length 5) of elements of R, such as (a₁, a₂, ..., a₅), where each a_i is an element of R.

The 5-tuples ( 1, 0,0,0,0), ( 0,1,0,0,0), (0,0,1,0,0), ( 0,0,0,1,0) and ( 0,0,0,0,1) form a basis for V.

Infinite dimensional Vector Spaces:

Example 1:

Consider the vector space P(R) of all polynomial functions on the real line. This vector space is not generated by any finite set. On the other hand, every polynomial is a finite linear combination of the polynomials f_n(x) = xⁿ , for n = 0, 1, 2, . . . , so these polynomials span P(R). Furthermore, if a polynomial _n xⁿ + _n-1 x^n-1 + · · · + ₁ x + ₀ is the zero function on R, then all coefficients _i must be zero, so the fn’s are linearly independent. In other words, the functions fn form a basis for the vector space P(R).

Example 2:

Let R be the vector space of infinite sequences (₁, ₂, ₃, . . . ,) of real numbers. This is a natural generalization of Rⁿ. The vector addition and scalar multiplication are defined in the natural way: the sum of (₁, ₂, ₃, . . .) and (₁, ₂, ₃, . . .) is (₁ +₁, ₂ +₂, ₃ +₃, . . .) and the product of (₁, ₂, ₃, . . .) by a scalar is the sequence (₁, ₂, ₃, . . .). Let V be the subspace of R spanned by the e_i for i = 1, 2, 3, . . . . Since the e_i are linearly independent, clearly they form a basis for V . The space V consists of all vectors in R in which all but finitely many of the slots are zeroes. This can be interpreted as the union of all Rⁿ as n goes to infinity.