Definition: The dimension of a vector space \(V\), denoted \(\dim(V)\), is the number of vectors in any basis of \(V\).
If \(V\) has a finite basis, it is called finite-dimensional. Otherwise, it is infinite-dimensional.
Examples:
- \(\dim(\mathbb{R}^n) = n\) (standard basis has \(n\) vectors)
- \(\dim(\mathbb{C}^n) = n\) over \(\mathbb{C}\)
- \(\dim(P_n) = n+1\) where \(P_n\) is the space of polynomials of degree at most \(n\) (basis: \(1, x, x^2, \ldots, x^n\))
- \(\dim(M_{m \times n}) = mn\) for \(m \times n\) matrices
- \(\dim(\{\mathbf{0}\}) = 0\)
Key theorems:
- If \(\dim(V) = n\), any set of \(n\) linearly independent vectors is a basis
- If \(\dim(V) = n\), any spanning set of \(n\) vectors is a basis
- If \(W\) is a subspace of \(V\), then \(\dim(W) \leq \dim(V)\)
- Rank-nullity theorem: For a linear transformation \(T: V \to W\), \(\dim(V) = \dim(\ker T) + \dim(\text{im } T)\)
Parents
Children
- Definition of dimension of a vector space (or subspace)The dimension of a finite-dimensional vector space is the number of vectors in any basis. All bases have the same...
- If a vector space has dimension nIn an $n$-dimensional space, every basis has $n$ vectors, every independent list has at most $n$, and every spanning...
- Every basis for a vector space contains the same number of elementsAll finite bases of a vector space have the same size. The exchange or dimension theorem shows an independent list...
- Definition of dimension of a vector space (or subspace) being finite or infiniteA vector space is finite-dimensional if some finite list spans it; its basis then has a finite number of elements....
- The dimension of a subspace is less than or equal to the dimension of the whole spaceFor $U\subseteq V$ with $V$ finite-dimensional, $\dim U\le\dim V$. A basis of $U$ is independent in $V$ and can be...
- Nested finite-dimensional subspaces of equal dimension are equalIf $U\subseteq W$ are finite-dimensional and $\dim U=\dim W$, then $U=W$. A basis of $U$ is an independent list in...
- A set of vectors containing more elements than the dimension of the space must be linearly dependentAny list longer than $\dim V=n$ in a finite-dimensional space is dependent. An independent list can have at most $n$...
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