📐 Unique Representation

If $\\{v_1, \\ldots, v_n\\}$ is a basis for $V$, then every vector $v \\in V$ can be written uniquely as $v = c_1 v_1 + \\cdots + c_n v_n$.

Proof: **Existence:** Since the basis spans $V$, every $v$ is a linear combination.

**Uniqueness:** Suppose $v = \\sum c_i v_i = \\sum d_i v_i$. Then:

$$\\sum (c_i - d_i) v_i = \\mathbf{0}$$

By linear independence, $c_i - d_i = 0$ for all $i$, so $c_i = d_i$.

From: Advanced Linear Algebra

Learn more: https://advanced-linalg.vercel.app/#/section/4

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