Basis for a Vector Space

Basis for a Vector Space

In a geometric space E, a linearly independent set of vectors such that every vector in E can be written as a linear combination of the vectors.

Through the linear combination of two vectors \(\overrightarrow{u}\) and \(\overrightarrow{v}\) with different directions, you can obtain all the vectors in the plane. Therefore, \((\overrightarrow{u},\space \overrightarrow{v})\) is said to be a basis for the set of vectors.

In other words, a basis for a vector space is a set of vectors that generates a vector space.


Consider the vectors \(\overrightarrow{u}\), \(\overrightarrow{v}\) and \(\overrightarrow{w}\).

The sides of the parallelogram ABCD are scalar multiples of \(\overrightarrow{u}\) and \(\overrightarrow{v}\). The construction below shows that: \(\overrightarrow{w} \space = \space a \overrightarrow{u} \space + \space b \overrightarrow{v}\).

basis vector space

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