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Linear combinations

Created over 8 years ago, updated 10 days ago

Definition: A linear combination of vectors (\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k) in a vector space (V) is any vector of the form:
[c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k]
where (c_1, c_2, \ldots, c_k) are scalars (from the underlying field).

The scalars (c_i) are called the coefficients of the linear combination.

Example: In (\mathbb{R}^3), the vector ((7, 4, -3)) is a linear combination of the standard basis vectors:
[(7, 4, -3) = 7(1,0,0) + 4(0,1,0) + (-3)(0,0,1)]

Example: The vector ((5, 8)) is a linear combination of ((1, 2)) and ((3, 4)) because:
[(5, 8) = 1 \cdot (1, 2) + 2 \cdot (3, 4)]

Linear combinations are the fundamental operation in linear algebra; they describe how vectors can be built from others using only addition and scalar multiplication.