definition of linearly independent

Nov 2009
75
0
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 like
Linear Algebra: The Basics

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


thanks!

 

Defunkt

MHF Hall of Honor
Aug 2009
976
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Israel
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 like
Linear Algebra: The Basics

give 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 \(\displaystyle a_1,a_2,...,a_n \in F\)
 
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dwsmith

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Mar 2010
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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.
 
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Nov 2009
75
0
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