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.
**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.