# Bases

Put content here**Definition:** A *basis* of a vector space \(V\) is a set of vectors \(\{\mathbf{v}_1, \ldots, \mathbf{v}_k\}\) that is:
1. **Linearly independent**, and
2. **Spans** \(V\) (i.e., \(\text{span}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\} = V\))
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Equivalently, a basis is a *minimal* spanning set or a *maximal* linearly independent set.
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**Standard basis of \(\mathbb{R}^n\):**
\[\mathbf{e}_1 = (1,0,\ldots,0), \quad \mathbf{e}_2 = (0,1,\ldots,0), \quad \ldots, \quad \mathbf{e}_n = (0,0,\ldots,1)\]
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**Example:** In \(\mathbb{R}^2\), the set \(\{(1,0), (0,1)\}\) is the standard basis. But \(\{(1,1), (1,-1)\}\) is also a basis.
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**Key theorem:** All bases of a given vector space have the same number of elements. This number is called the *dimension* of the space.
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**Coordinates:** Once a basis is chosen, every vector \(\mathbf{v} \in V\) can be written *uniquely* as \(\mathbf{v} = c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k\). The coefficients \((c_1, \ldots, c_k)\) are the coordinates of \(\mathbf{v}\) with respect to that basis.

# Parents

* AbstractCoordinate vector spaces
* CoordinateAbstract vector spaces