# definition of linearly independent

#### matlabnoob

hello

can anyone tell me if this definition is correct for linearly independent
..

'A subset S of a vector space V is called linearly dependent if there exists a finite number of distinct vectors v1, v2, ..., vn in S and scalars a1, a2, ..., an, not all zero, such that '
that is from wikipedia! ( Linear independence - Wikipedia, the free encyclopedia )

i thought a1, a2, ..., an = 0

??

some sites likegive a different definition!! which one is true? whose word do i take? (Thinking)

thanks!

#### Defunkt

MHF Hall of Honor
hello

can anyone tell me if this definition is correct for linearly independent
..

'A subset S of a vector space V is called linearly dependent if there exists a finite number of distinct vectors v1, v2, ..., vn in S and scalars a1, a2, ..., an, not all zero, such that '
that is from wikipedia! ( Linear independence - Wikipedia, the free encyclopedia )

i thought a1, a2, ..., an = 0

??

some sites likegive a different definition!! which one is true? whose word do i take? (Thinking)

thanks!

As you can see, and as you quoted yourself, the definition you gave is for linear dependency. The definition for linear independency is that $$\displaystyle \{v_1,...,v_n\}$$ are linearly independent over a field $$\displaystyle F$$ if:

$$\displaystyle a_1v_1 + ... + a_nv_n = 0 \Leftrightarrow a_1=a_2=...=a_n=0$$ for scalars $$\displaystyle a_1,a_2,...,a_n \in F$$

• matlabnoob

#### dwsmith

MHF Hall of Honor
A very simple way to view linear independence is that none of the vectors can be written as a linear combination of each other.

Example: $$\displaystyle \mathbb{R}^2$$

There are only two 2 linear independent vectors in $$\displaystyle \mathbb{R}^2$$ (which two you choose is arbitrary as long as they are orthogonal). If the you are giving 3 vectors, one of the vectors has to be a linear combination of the two.

Example: $$\displaystyle P_5$$ all polynomials of less than degree 5 (some books refer to $$\displaystyle P_5$$ as degree 5 or less) but in my example it is strictly less than 5.

If you are giving 6 polynomials, by the Pigeon Hole Principle, one of them has to be linear dependent since 5 polynomials span the space giving.

Whenever you have more vectors than the dimension, you will have linear dependence of at least one vector.

• matlabnoob

#### matlabnoob

A very simple way to view linear independence is that none of the vectors can be written as a linear combination of each other.

Example: $$\displaystyle \mathbb{R}^2$$

There are only two 2 linear independent vectors in $$\displaystyle \mathbb{R}$$ (which two you choose is arbitrary). If the you are giving 3 vectors, one of the vectors has to be a linear combination of the two.

Example: $$\displaystyle P_5$$ all polynomials of less than degree 5 (some books refer to $$\displaystyle P_5$$ as degree 5 or less) but in my example it is strictly less than 5.

If you are giving 6 polynomials, by the Pigeon Hole Principle, one of them has to be linear dependent since 5 polynomials span the space giving.

Whenever you have more vectors than the dimension, you will have linear dependence of at least one vector.

thanks! ill note this for future