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I have a very hard time remembering which is which between linear independence and linear dependence... that is, if I am asked to specify whether a set of vectors are linearly dependent or independent, I'd be able to find out if $\vec{x}=\vec{0}$ is the only solution to $A\vec{x}=\vec{0}$, but I would be stuck guessing whether that means that the vectors are linearly dependent or independent.

Is there a way that I can understand what the consequence of this trait is so that I can confidently answer such a question on a test?

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7 Answers 7

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Intuitively vectors being linearly independent means they represent independent directions in your vector spaces, while linearly dependent vectors means they don't. So for example if you have a set of vector $\{x_1, ..., x_5\}$ and you can walk some distance in the $x_1$ direction, then a difference distance in $x_2$, then again in the direction of $x_3$. If in the end you are back where you started then the vectors are linearly dependent (notice that I did not use all the vectors).

This is the intuition behind the notion and you can make it into a definition because in the above example if we start at $0$ then we walk $a_i$ in the $x_i$ direction, then the above paragraph says that $a_1x_1+a_2x_2+a_3x_3=0$. (This is how you should think of linear combinations, as directions to go given by your vectors.)

Finally I will say that you should memorize the definitions. I've taught linear algebra to students that it is their first proof-based math class, and many students don't realize how important knowing the PRECISE definition is. Definitions are crucial and changing one single word can completely change the meaning. So my advice when just starting out is that you should make flash cards of ALL definitions in your book and memorize them. Then once you know them exactly look at the examples after the definition in the book and see how the examples fit the definition.

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The vectors are dependent ('they depend on one another') if there is some relation among them (in addition to the one with all 0 present for any collection of vectors). So, dependent means there is some relation other than all 0.

Differently: independent means if you want a linear combination of the vectors to sum to the 0 vector, you need to assure that each part of the coombination independently is 0; thus each coordinate in the solution $0$.

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A broader perspective on linear dependence is the theory of relations in group theory. Roughly speaking, a relation is some equation satisfied by the elements of a group, e.g. $(ab)^{-1}=b^{-1}a^{-1}$; relations basically amount to declaring how group elements depend on each other. One useful convenience is that relations can always be put into the form "$\rm blah=identity~element$" by simply inverting one side over to the other, e.g. $ab=c\Leftrightarrow abc^{-1}=e$.

Abelian groups (generally, modules) have additive group operations, so a relation would look like an equation $2a+b=3c$ or equivalently $2a+b-3c=0$. In particular a vector space is a module over a field, so instead of just integers we can have any field scalars involved in our equations with vectors. Ultimately, a linear dependency is where vectors satisfy some relationship with each other.

Conversely, a set of vectors is linearly independent if they satisfy no linearity equation other than the obvious, trivial one involving only zeros (this case is uninteresting because it applies universally and so essentially says nothing of value). So e.g. $2a+b=3c$ is impossible if $\{a,b,c\}$ is L.I.

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    $\begingroup$ You went "Bill Dubuque" on this one, all the way! Quite enlightening. +1 $\endgroup$
    – Pedro
    Jul 31, 2013 at 0:06
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I've found the best way to understand this is as follows: a set of vectors is linearly dependent if you can write one of them in terms of the others.

When you multiply a matrix by a vector, $A\vec{x}$, that's shorthand for "multiply each column of $A$ by the corresponding entry of $\vec{x}$, and then add them together." If the columns in $A$ are linearly dependent, then there's some $\vec{x}$ that will allow them to cancel, giving you the zero vector.

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It is a long answer but kindly bear with me

To understand linear dependence and linear independence we first need to understand linear combination and span. I assume only two in a 2D plane.

Span

The span of two vectors v1 and v2 is the set of all their linear combinations.

OR

The set of all possible vectors which can be reach through the linear combination of two vectors V1 and V2 is the span of those two vectors.

What Is The Span Of A Single Vector

The span of a single vector is all the vectors which lie on the single line.

Linear Dependence

Lets say we have two vectors in a 2D plane and they are collinear that is one of the vector is redundant. It means one of the vector is not adding anything to the span of the first vector. In such case the two vectors are known as linearly dependent.

Mathematical Definition of Linear Dependence

Let S be the set of vectors S = {V1, V2, V3,…..,Vn} The set S is linearly dependent if and only if CV1+ C2V2 + C3V3 +….+ CnVn=zero vector for some all Ci’s at least one is non zero. The condition of checking linear dependence if c1 or c2 is non zero then the two vectors are linearly dependent

Linearly Independence

If in a 2D plane the two vectors V1 and V2 are not collinear then one of the vector is increasing the span of the first vector that is with only vector the span was just a single line but with the linear combination of V1 and V2 we can reach every single vector in the 2D plane(Span of V1 and V2 is the whole 2D plane). It means that no vector is redundant. In such case the two vectors are known as linearly independent.

Mathematical Definition of Linear Independence

Let S be the set of vectors S = {V1, V2, V3,…..,Vn} The set S is linearly independence if and only if CV1+ C2V2 + C3V3 +….+ CnVn=zero vector The condition of checking linear independence if c1 and c2 are both zero then the two vectors are linearly independent

But Why This Formulas Make Sense?

The conditions to check the linear dependence/independence basically check whether the two vectors in the 2D plane are collinear or not. Lets dive into it deeper.

We know that to find the linear combination of two vectors we multiply the vectors by some scalar and add them. Since we equated our linear combination of V1 and V2 to zero vector .It means we are basically asking the question that to reach the zero vector by the linear combination of V1 and V2 by which scalar we need to multiply our vectors. We got c1=c2=0 in our example that means the only way to reach the zero vector by the linear combination of V1 and V2 is to multiply those vectors by 0. This shows that the two vectors V1 and V2 do not lie on the same line and hence they are Linearly independent because the only way to reach the zero vector by the linear combination of V1 and V2 is to scale both the vectors by zero.

Note:

If V1 and V2 were collinear there will be infinite values of c1 and c2 through which we can reach the zero vector by the linear combination of two vectors.(The vectors will be opposite and direction having the same magnitude).

I assumed that we are working in 2D plane. The concept of Linear Dependence/Independence also applies to higher dimensions but this intuition of collinearity will not be applicable in higher dimensions.

I hope it helps.

Thanks for reading :)

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Consider three linearly independent vectors $X=(x,0,0),Y=(0,y,0),Z=(0,0,z)$ that make up a standard basis of $\mathbb{R^{3}}$. These vectors, by themselves, form the $xyz-axis$. If we took away the vector $(0,0,z)$, what linear combination of the vectors $X= (x,0,0),Y= (0,y,0)$, or $span(X,Y)= cX + kY$ would get us $Z$? None, because there is no combination $c(x,0,0)+ k(0,y,0) = (0,0,Z)$ as $Z \notin span(X,Y)$. This is because $Z$ or the $z-axis$ is linearly independent from the $x-axis$ and the $y-axis$ which fill out the $xy-plane$.

Note that this is not formally how linear independence is defined, just a fairly intuitive way to visualize a set of linearly independent vectors.

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This result can help you:

A subset $S$ of a vector space is linearly independent if and only if for any distinct $\vec s_{1}, ... , \vec s_{n} \in S $ the only linear relationship among those vectors

$$ c_{1} \vec s_{1} + ... + c_{n} \vec s_{n} = \vec 0$$

with $c_{1},...,c_{n} \in \mathbb{R}$ is the trivial one: $c_{1}=0,...,c_{n}=0$.

This is taken of Jim Hefferon's book Linear Algebra (http://joshua.smcvt.edu/linearalgebra/, Chapter Two, Section II, Lemma 1.3).

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