Basis of a Vector Space

Vector Space

Let's introduce some definitions. A set is called closed with respect to some operation if for any elements of the set the result of applying this operation belongs to the given set. A set of vectors that is closed under linear operations is called a vector space (or linear space).

For example, the set of all vectors parallel to a given plane is a two-dimensional vector space. In fact, if the vectors a, b are parallel to the plane π, then any linear combination of these vectors is also parallel to this plane:

Similarly, the set of all vectors parallel to a given straight line forms a one-dimensional vector space. The zero vector space consists only of the zero vector. A three-dimensional vector space includes the set of all vectors of the space.

More formally, for a set to be a vector space it must satisfy the 8 axioms listed below.

Let V be a non-empty set over a scalar field F. The elements of the set V are vectors a, b, c, ..., and the elements of the field F are numbers λ, μ, ν, ... Two binary operations are defined for the set V over field F:

• Vector addition:
  \[\forall \mathbf{a}, \mathbf{b} \in V \;\exists\, \mathbf{c} \in V: \mathbf{a} + \mathbf{b} = \mathbf{c};\]
• Scalar multiplication:
  \[\forall \mathbf{a} \in V,\;\forall \lambda \in F \;\exists\, \mathbf{b} \in V: \lambda\mathbf{a} = \mathbf{b}.\]

The set V is a vector space over F, if the following axioms are satisfied:

1. \(\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}\;\) (commutativity)
2. \(\left({\mathbf{a} + \mathbf{b}}\right) + \mathbf{c} = \mathbf{a} + \left({\mathbf{b} + \mathbf{c}}\right)\;\) (associativity)
3. \(\mathbf{a} + \mathbf{0} = \mathbf{a}\;\) (existence of identity element)
4. \(\mathbf{a} + \left({-\mathbf{a}}\right) = \mathbf{0}\;\) (existence of inverse element)
5. \(1 \cdot \mathbf{a} = \mathbf{a}\)
6. \(\lambda\left({\mu\mathbf{a}}\right) = \left({\lambda\mu}\right)\mathbf{a}\)
7. \(\lambda\left({\mathbf{a} + \mathbf{b}}\right) = \lambda\mathbf{a} + \lambda\mathbf{b}\)
8. \(\left({\lambda + \mu}\right)\mathbf{a} = \lambda\mathbf{a} + \mu\mathbf{a}\)

Here a, b, c, 0 are vectors and λ, μ, 1 are numbers.

Examples of Vector Spaces

A key example of a vector space is the set of all free vectors with respect to the usual operations of adding vectors and multiplying a vector by a real number. However, vector space is a rather abstract concept. It can contain not only vectors but also other objects. For example, the following sets are also vector spaces:

• \({M_{n \times m}} -\) the set of matrices of dimension \(n \times m\) with respect to the operation of matrix addition and multiplication by an element of the field of numbers F form a vector space.
• \({P_n\left({x}\right)} -\) the set of polynomials with real coefficients of degree not higher than n form a vector space with respect to the operation of addition of polynomials and multiplication of a polynomial by a real number.
• The set of all functions of the form \(f : \mathbb{R} \to \mathbb{R}\) is a vector space; the addition of vectors and their multiplication by numbers are given by the formulas
  \[\left({f + g}\right)\left({x}\right) = f\left({x}\right) + g\left({x}\right),\;\;\left({\alpha \cdot f}\right)\left({x}\right) = \alpha\cdot f\left({x}\right).\]
  where \(f\left({x}\right), g\left({x}\right)\) are functions, \(\alpha, x\) are real numbers.
• The set of all convergent sequences \(\{u_n\},\) \(\{v_n\}\) with the defined operations of addition and multiplication by a number is a vector space. The sum of sequences and the product of a sequence by a number are given by the formulas
  \[\{u_n + v_n\} = \{u_n\} + \{v_n\},\;\;\{\alpha \cdot u_n\} = \alpha \cdot \{u_n\}.\]

From the above examples it can be seen that the elements of a vector space can be of completely different nature: geometric vectors, matrices, polynomials, functions, sequences, and so on.

Basis of a Vector Space

Three linearly independent vectors a, b and c are said to form a basis in space if any vector d can be represented as some linear combination of the vectors a, b and c, that is, if for any vector d there exist real numbers λ, μ, ν such that

This equality is usually called the expansion of the vector d relative to the basis a, b, c and the numbers λ, μ and ν are called the coordinates of the vector d with respect to the basis a, b, c.

Every vector d can be decomposed in a unique way in terms of the basis a, b, c.

A basis for a plane is defined similarly. Two linearly independent vectors a and b lying in a plane π are said to form a basis in this plane if any vector c lying in the plane π can be represented as some linear combination of the vectors a and b, that is, if for any vector c lying in the plane π there exist real numbers λ and μ such that

The following statements are true:

1. Any triple of non-coplanar vectors a, b and c form a basis in space.
2. Any pair of non-collinear vectors a and b lying in a given plane form a basis in this plane.

Solved Problems

Solution.

Two vectors form a basis in the plane if they are not collinear. Since collinear vectors satisfy the relation \(\mathbf{a} = \lambda\mathbf{b},\) their coordinates must be proportional, that is, the determinant composed of the coordinates of these vectors must be equal to zero. Check it out:

Hence, the vectors \(\mathbf{a}\) and \(\mathbf{b}\) are non-collinear and form a basis. The coordinates of the vector \(\mathbf{c}\left({-7,5}\right)\) in this basis are determined from the equation

Write it in coordinate form

So the vector \(\mathbf{c}\) in basis \(\left({\mathbf{a, b}}\right)\) has the coordinates \(\left({\lambda,\mu}\right) = \left({2,1}\right).\)

Solution.

For three vectors \(\mathbf{a},\) \(\mathbf{b}\) and \(\mathbf{c}\) to form a basis in space, they must be non-coplanar. Recall that the conditions for complanarity are

If you make a matrix of three coplanar vectors, its determinant will be equal to zero since the third vector \(\mathbf{c}\) linearly depends on the vectors \(\mathbf{a}\) and \(\mathbf{b}.\) Accordingly, the matrix of three non-coplanar vectors has a non-zero determinant. Let's check what we have in our case:

Therefore, the vectors \(\mathbf{a},\) \(\mathbf{b},\) \(\mathbf{c}\) are linearly independent and form a basis in space. Find the coordinates of the vector \(\mathbf{d}\) in this basis. Write the expansion of this vector in the basis:

Go to the coordinate form:

Therefore, the coordinates of the vector \(\mathbf{d}\) in this basis are \(\left({\lambda,\mu,\nu}\right) = \left({1,0,2}\right).\)

Solution.

Express the vector \(\mathbf{BD}\) in terms of vectors \(\mathbf{AB} = \mathbf{a}\) and \(\mathbf{AD} = \mathbf{b}.\) By the vector subtraction rule,

Find now the vector \(\mathbf{CO}.\) Notice that

Since \(\mathbf{CA} = - \mathbf{AC}\) and \(\mathbf{CO} = \frac{1}{2}\mathbf{CA},\) we get

or

Calculate the vector \(\mathbf{MD}:\)

where

Hence

Solution.

As a basis we can take the following matrices

Then an arbitrary \(2 \times 2\) matrix is represented in this basis as follows:

In our case we have

This matrix in the given basis has coordinates \(\left({-1,5,3,0}\right).\)