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# Dimension

Put content here.**Definition:** The *dimension* of a vector space \(V\), denoted \(\dim(V)\), is the number of vectors in any basis of \(V\).
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If \(V\) has a finite basis, it is called *finite-dimensional*. Otherwise, it is *infinite-dimensional*.
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**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
- \(\dim(\{\mathbf{0}\}) = 0\)
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**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

* Coordinate vector spaces