Dimensions of vector space
Kapitoly: Vector spaces, Examples of vector spaces, Vector subspace, Linear combinations of vectors, Linear wrapper, Bases of vector space, Dimensions of vector space, Transition matrix
The dimension of a vector space is equal to the number of elements of the basis of that vector space. If the basis is infinite, then we say the dimension is infinite.
Definition of the dimension of a vector space
Consider a vector space V and some basis B. That is, B ⊆ V, the set B contains linearly independent vectors and the linear cover of the set B is equal to the space V: <B> = V.
If the basis B has a finite number of elements n, then we say that the dimension of the space V is n. We denote the dimension by dim, so dim V = |B| = n. If the basis B is infinite, we say that the dimension of the space is also infinite. Since all bases of space are always the same size, it doesn't matter which base we take.
Example: the space ℝ3 has the basis [1,0,0], [0,1,0], [0,0,1]. There are three vectors in total, so $\dim \mathbb{R}^3 = 3$. In general, we can say that $\dim \mathbb{R}^n = n$.
If the space V has dimension n, we talk about n-dimensional vector space.
Properties of the dimension of a vector space
- Consider a vector space V and some subspace of W ⊆ V. Then $\dim W \le \dim V$.
- Given a vector space V and some subspace of W ⊂ V. Then it holds that $\dim W < \dim V$. This point is different in that we do not allow W = V. Thus, if we have a subspace W that is smaller than V, then this subspace will also have a lower dimension. Why? Because there is a vector x ∈ V ∖ W, which is not a linear combination of the vectors from W, and hence not a linear combination of the vectors from any basis W.
- If the space V has dimension n, then any n linearly independent vectors form the basis of the space V. The set of n + 1 vectors then forms a linearly dependent set of vectors.