Linear Dependence - Vectors

[danielf]

I've been tught a different method to the VCAA/NEAP/Exambusters way
If you have three vectors a, b, and c, then they are linearly dependent if ma + nb = c where m and n are non-zero real number coefficients
VCAA says that to prove linear dependence, you must prove that ma + nb + lc = 0 where m,n,l don't equal zero
I realise that its essentially the same thing but will i get marked down for my method?
Also, with both bethods, can one of the coefficients equal zero (the other being a real number) and the vectors still be linearly dependent?

[hanhanchampion]

good question......  But if one of the coefficients is zero, wouldn't the other two vectors equal to each other cause of the linear dependence?

[Mao]

the two methods are equivalent.

and if one of the coefficient is zero, the set of vectors would be linearly dependent.
another thing is that if one of the vectors is a zero vector, the set would also be linearly dependant,

A geographic example may help to clarify the concept of linear independence. A person describing the location of a certain place might say, "It is 5 miles north and 6 miles east of here." This is sufficient information to describe the location, because the geographic coordinate system may be considered a 2-dimensional vector space (ignoring altitude). The person might add, "The place is 7.81 miles northeast of here." Although this last statement is true, it is not necessary.
In this example the "5 miles north" vector and the "6 miles east" vector are linearly independent. That is to say, the north vector cannot be described in terms of the east vector, and vice versa. The third "7.81 miles northeast" vector is a linear combination of the other two vectors, and it makes the set of vectors linearly dependent, that is, one of the three vectors is unnecessary.

taking that definition on board, the prior two statements can be seen to be true


the common misconception with linear dependency is that the vectors can form an enclosed polygon [or the vectors are parallel]. This is true in some cases, but not always true. It complies with the above geometric meaning, that one or more vectors in that set can be defined in terms of the others hence unnecessary. However, it does not allow for the two special cases.

[rowshan]

Im not sure what the wikipedian folk are on about...
"linearly dependent, that is, one of the three vectors is unnecessary"
why is one of the three vectors unnecessary?

"the common misconception with linear dependency is that the vectors can form an enclosed polygon [or the vectors are parallel]. "
HUh!? why is this a misconception? What are the two special cases?

[Collin Li]

Im not sure what the wikipedian folk are on about...
"linearly dependent, that is, one of the three vectors is unnecessary"
why is one of the three vectors unnecessary?

Because if they are dependent, any of the other two vectors can form a linear combination to produce the third vector, hence making it "unnecessary."

[Collin Li]

"the common misconception with linear dependency is that the vectors can form an enclosed polygon [or the vectors are parallel]. "
HUh!? why is this a misconception? What are the two special cases?

A linear combination of the vectors can form an enclosed polygon, but it is not necessarily true that the vectors alone can form an enclosed polygon.

The vectors need not necessarily be parallel in three dimensions or above.

[Mao]

a set of vectors is linearly dependent when one or more of the vectors is "unnecessary", as it can be defined in terms of the other vectors.

i.e. are linearly independent, as none of those three can be defined in terms of each other.
however, would be linearly dependent, as the final vector can be [and is] written in terms of the other vectors, hence is unnecessary.

another example is , this set is linearly dependent, as the second vector can be written in terms of the first. However, this set of vectors will NOT form an enclosed polygon [try it yourself], hence the misconception.
the other special case is (any set that includes one or more zero vectors). the zero vector is unnecessary as it doesn't give any information [there are more pedantic ways of saying this].

this becomes useful when you need to describe one thing in terms of another. for example, you would describe a vector , but not , as the vectors are linearly dependent, and some of those vectors are unnecessary as the information they convey can be expressed using other vectors in that set.

[fredrick]

so is this set linearly dependant? cause it can be written in terms of the others?

[Mao]

yep

[rowshan]

to prove linear dependence i have show that for ma + nb + lc = 0, m,n,l are not zero.
but that example that ya provided(mao) (2i,-2i,j) would not satisfy the above now would it?
That is m,n,l have to equal for the equation to be satisfied; therefore this set is linearly independent?

[rowshan]

Coblin, i get the necessary thing lol, thanks. I'm a little confused with the "vectors alone" bit. Do you mean that these vectors, without any urmmm dilations(?), does not necessarily have for a fully enclosed polygon?
But it does have to make a fully enclosed polygon with the dilations right?

[Mao]

ummm no.

one set of m,n,l would be {1,-1,0}, i can also use {-1,1,0}, or {-50000,50000,0}, they all work

m,n,l can be anything, so long as not all are zero [i.e. not {0,0,0}]

[Mao]

Coblin, i get the necessary thing lol, thanks. I'm a little confused with the "vectors alone" bit. Do you mean that these vectors, without any urmmm dilations(?), does not necessarily have for a fully enclosed polygon?
But it does have to make a fully enclosed polygon with the dilations right?

no it doesn't have to make a fully enclosed polygon with "dilations". as are the two cases I've shown involving pairs of parallel vectors and the zero vector

[danielf]

thanks heaps fellas geniusi all of you

[rowshan]

"another example is { 2{i},-2{i},{j}}, this set is linearly dependent, as the second vector can be written in terms of the first."
so this means that not all the vectors in the set have to be able to be expressed in terms of the other?

If you were asked to prove that the set {-1,1,0}, or {-50000,50000,0} is linearly dependent how would you do it?

thanks for the help =).