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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: Yitzi_K on September 12, 2010, 12:24:53 pm

Title: Linear dependence/independence of vectors
Post by: Yitzi_K on September 12, 2010, 12:24:53 pm: Could someone please explain this topic to me? My teacher didn't do the best job, and I'm somewhat confused about the whole thing.
Thanks
Title: Re: Linear dependence/independence of vectors
Post by: f.sharp on September 12, 2010, 12:45:06 pm: if 2 vectors can be used to represent a 3rd, they are linearly dependent. say 3 vectors that form a triangle, the sum of all 3 equals 0. if not then theyre independent.

oh and if u have 2 vectors, they are almost always independent UNLESS one is the scalar multiple of the other.
Title: Re: Linear dependence/independence of vectors
Post by: Martoman on September 12, 2010, 04:29:57 pm: Quote from: f.sharp on September 12, 2010, 12:45:06 pm
if 2 vectors can be used to represent a 3rd, they are linearly dependent. say 3 vectors that form a triangle, the sum of all 3 equals 0. if not then theyre independent.

oh and if u have 2 vectors, they are almost always independent UNLESS one is the scalar multiple of the other.

Great explanation. The idea of forming a triangle with three vectors is how I see it. With two vectors only if they are parallel they are linearly dependent. A trivial proof of this is that:

$a=kb$ ie b is parallel to a, then $a-kb = 0$ ie they will always form a linearly dependent set.
Title: Re: Linear dependence/independence of vectors
Post by: Yitzi_K on September 12, 2010, 05:37:27 pm: Quote from: f.sharp on September 12, 2010, 12:45:06 pm
if 2 vectors can be used to represent a 3rd, they are linearly dependent. say 3 vectors that form a triangle, the sum of all 3 equals 0. if not then theyre independent.

oh and if u have 2 vectors, they are almost always independent UNLESS one is the scalar multiple of the other.

When you say that the 3 vectors added together equals 0, is that a + b + c = 0, or ka + pb + qc= 0?
Title: Re: Linear dependence/independence of vectors
Post by: jasoN- on September 12, 2010, 05:39:13 pm: its the latter, I should know, got it wrong in a recent sac costing me full marks (i used a + b + c = 0)

To add instead of using ka + pb + qc= 0
just use a = pb + qc, the real coefficients will be different but its more simple to get an answer
Title: Re: Linear dependence/independence of vectors
Post by: m@tty on September 12, 2010, 07:05:15 pm: Quote from: jasoN- on September 12, 2010, 05:39:13 pm
To add instead of using ka + pb + qc= 0
just use a = pb + qc, the real coefficients will be different but its more simple to get an answer

Definitely.

Also worthy of note that three vectors in two dimensions are ALWAYS dependent. Same with 4 in 3 dimension, n+1 vectors in n dimensions.
Title: Re: Linear dependence/independence of vectors
Post by: Mao on September 13, 2010, 06:54:15 pm: A quick note on the interpretation:

A set of vectors are linearly independent if no vectors in the set can be represented by a linear combination of other vectors in the same set. That is, if a set of vectors {a, b, c, d} is linearly independent, then a cannot be expressed in terms of b, c and d, and b cannot be expressed in terms of a, c and d, and so on.

Independent sets are often useful to describe a 'span'. A 2D plane can be completely spanned by the two vectors $\{ i, j \}$ . That is to say, any vectors in the 2D plane can be described by a linear combination of these two.

If you have a set of vectors $\{ v_1, v_2, v_3 \} = \{ i, i + j, 2i + 3j \}$ , this set of vectors is not linearly independent because $v_3 = 3v_2 - v_1$ . That is, $\{ v_1, v_2 \}$ spans all linear combinations of $\{ v_1, v_2, v_3 \}$ . In this sense, $v_3$ is 'redundant'.
Title: Re: Linear dependence/independence of vectors
Post by: Yitzi_K on September 13, 2010, 10:10:33 pm: Thanks a lot guys this has been a real help. :)