# Basis

It is said that the vector *w* is a **linear combination** of other vectors, *u*_{1},*u _{2},…,u_{n}*, if there are k

_{1},k

_{2},…,k

_{n}ЄR, such that:

*w*= k

_{1}

*u*+ k

_{1}_{2}

*u*+….+ k

_{2}_{n}

*u*

_{n}

Example: *w*(3,-3) is linear combination of *u*(1,1) and *v*(0,3), because (3,-3)=3(1,1)-2(0,3)

A set of vectors is said to be** linearly dependent** if one of them is linear combination of the others. Otherwise, the set is said to be **linearly independent**.

In the plane, two vectors are linearly dependent if and only if they are proportional.

Example: *u*(1,3),*v*(-3,-9),*w*(1,0); *u* and *v* are linearly dependent; *u* and *w* are linearly independent.

A set of vectors is said to be a **generative system** if all vectors can be expressed as linear combination of these vectors.

Example: {(1,0),(1,1),(0,1)} is generative system of the plane.

A** basis** is a set of vectors linearly independent and generative system. If the elements are orthogonal and their magnitudes are 1 (unit vectors), the basis is called **orthonormal basis**.

Example: B_{C}={i(1,0),j(0,1)} is an orthonormal basis called **canonical basis**.

**Exercise**

Let *S*={*u _{1}(1,1),u_{2}(1,0),u_{3}(-1,-1)*}. Decide if the following sentences are true or false:

a) *S* is a generative system

b) *S* is a basis

c) *u*_{1} and *u*_{3} are linearly independent

d) *u*_{1} and *u*_{2} form a basis

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