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I am not very proficient with mathematics. I have the following question.

When do we say two vectors are linearly independent, in X-Y coordinate system (2-D case)?

I read that any two non parallel vectors are linearly independent in 2-D space.

But from various other contexts I find that the necessary and sufficient condition for two vectors to be linearly independent is that they must be orthogonal. And, this appeals to me to be true conceptually. I am interested only in two dimensional vectors.

Any clarification in this regard would be highly appreciated.

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  • $\begingroup$ Well two vectors $v_1, v_2$ are linearly independent if $av_1 + bv_2 = 0$ is only satisfied when both $a,b$ are zero. But to say that they are linearly dependent means that there are scalars $a,b$ not all zero so that $av_1 + bv_2 = 0$. If $a \neq 0$, then $v_1 = -\frac{bv_2}{a}$. Or if $b$ is non-zero, you can deduce one is the multiple of another. This is the definition of linear (in)dependence, so that's why you get two vectors are linearly independent if they're not a multiple of each other. $\endgroup$ – MathNewbie Jun 8 '15 at 7:34
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A vector (in 2-D) is said to be linearly dependent when one can be obtained from the other by multiplying by a scalar or a number. For example, $(1,0)$ and $(3,0)$ are linearly dependent because you can obtain latter by multiplying former by 3 or obtain former by multiplying latter by $\frac{1}{3}$.

So, they are linearly independent when they are not linearly dependent.

By formal definition, a subset $S$ of a vector space $V$ is said to be linearly dependent if there exist a finite number of distinct vectors, $x_1, x_2, ..., x_n$ in $S$ and scalars $a_1, a_2, ..., a_n$ not all zero, such that $a_1x_1+a_2x_2+...+a_nx_n = 0$. In this case, S is said to be linearly dependent.

A subset $S$ of a vector space $V$ is said to be linearly independent if $S$ is not linearly dependent.

So, we can also say that a set of vectors (in any dimension) are linearly dependent if a vector in it can be written as linear combination of the other vectors in set. Whereas they are linearly independent if this statement does not hold.

I don't know how much you have studied linear algebra so far, but if you are interested you can study from Linear Algebra by Gilbert Strang (MIT OCW has Strang's lectures as well). It is good for beginners and Linear Algebra is very interesting subject. Good Luck!

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  • $\begingroup$ My problem arose when I saw Maxwell was insistent on orthogonality of two vectors so that they are independent, in deriving his equation of distribution of speeds of molecules in an ideal gas at equilibrium.Why should he insist on orthogonality? $\endgroup$ – Radhakrishnamurty P Jun 9 '15 at 12:19
  • $\begingroup$ When two vectors are orthogonal, it geometrically means that they are perpendicular to each other. So, no vector is parallel to the other one, which further implies that one can't be obtained by multiplying the other by a scalar (remember - scalar multiplication to vector only changes magnitude not direction). Thus, the two vectors will be linearly independent... About the derivation by Maxwell, I didn't study about equation of distribution of speed of molecules, as I only studied pure math and applied math, but I hope you are able to understand how orthogonality implies linear independence... $\endgroup$ – Ritu Jun 10 '15 at 15:47
  • $\begingroup$ I understand that orthogonal vectors are linearly independent and my conviction is that it is a necessary and sufficient condition for linear independence. (Maxwell stresses on orthogonality in his derivation. Let us not delve further into that but concentrate only on linear dependence). $\endgroup$ – Radhakrishnamurty P Jun 10 '15 at 15:55
  • $\begingroup$ I understand that orthogonal vectors are linearly independent and my conviction is that it is a necessary and sufficient condition for linear independence. (Maxwell stresses on orthogonality in his derivation. Let us not delve further into that but concentrate only on linear dependence). A set of 3 or more vectors that form a loop add up to a zero vector. Any of the component vectors say Vi and the zero vector are linearly independent, since the sum of a non zero scalar multiple of Vi and the zero vector is a non zero vector. But, can Vi and sigma Vi be linearly independent? $\endgroup$ – Radhakrishnamurty P Jun 10 '15 at 16:22
  • $\begingroup$ I am sorry, I did not get what you are trying to ask... Can you explain it again? Also use Latex to write mathematical symbols and equations (You may refer detexify.kirelabs.org/classify.html)... $\endgroup$ – Ritu Jun 12 '15 at 7:39

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