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.
Children
- Definition of linear combination of vectorsA linear combination of vectors $v 1,\ldots,v k$ over $\mathbb F$ is a vector $c 1v 1+\cdots+c kv k$ with scalars $c...
- Example of linear combination of vectors in R^2Let $u=(1,0)$ and $v=(1,1)$. Then $2u+3v=(5,3)$. Conversely, $(a,b)=(a-b)u+bv$, so these vectors span $\mathbb R^2$....
- Definition of weights in a linear combination of vectorsThe weights of a linear combination $\sum i c iv i$ are its scalar coefficients $c i$. In $2u-3v$, the weights are...
- Example of writing a given vector in R^3 as a linear combination of given vectorsLet $v 1=(1,0,1)$, $v 2=(0,1,1)$ and $b=(2,3,5)$. Solving $c 1v 1+c 2v 2=b$ gives $c 1=2$ from the first coordinate...
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