Definition: A set of vectors \(\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k\}\) is linearly dependent if there exist scalars \(c_1, \ldots, c_k\), not all zero, such that:
\[c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k = \mathbf{0}\]
The set is linearly independent if the only solution to this equation is \(c_1 = c_2 = \cdots = c_k = 0\).
Intuition: Linear dependence means at least one vector can be written as a linear combination of the others. Linear independence means no vector is redundant.
Example (dependent): In \(\mathbb{R}^2\), the vectors \((1,2)\), \((2,4)\) are dependent because \(2(1,2) + (-1)(2,4) = (0,0)\).
Example (independent): The standard basis vectors \((1,0)\), \((0,1)\) in \(\mathbb{R}^2\) are independent: the only way \(c_1(1,0) + c_2(0,1) = (0,0)\) is if \(c_1 = c_2 = 0\).
Key fact: Any set containing the zero vector is linearly dependent. Any set with more vectors than the dimension of the space is linearly dependent.
Children
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