Linear Combination

up:: Vector Space

Given a vector space $(V,+,\cdot )$ and a subset $\{v_i\}\subset V$, we say that a vector $v$ is a linear combination of $\{v_i\}$ if there are coefficients $\{{\alpha }_i\}$ in the vector space’s Field such that

$$v=\sum _i{\alpha }_i{v}_i$$

That is, $v$ can be decomposed into components of the vectors $\{v_i\}$.

Related

• A subset of a vector space is Linearly Dependent if there is at least one vector inside it which is a linear combination of the other vectors
• A subset is Linearly Independent if no vectors inside it are linear combinations of any other vectors.

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References

• Um Curso de Álgebra Linear, Flávio Ulhoa Coelho & Mary Lilian Lourenço. Editora EDUSP.