Linear Independence: A Guide to Determining Vectors' Dependence"

[Gregg]

linear independence

Homework Statement

given that a,b and c are linearly independant vectors determine if the following vectors are linearly independant.

a) a,0

b) a+b, b+c, c+a

c) a+2b+c, a-b-c, 5a+b-c

The Attempt at a Solution

I'm not sure how to tackle the question in this form.

Edit:

a) a=0a Dependant

$\text{Det}\left[\left( \begin{array}{ccc} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{array} \right)\right]=2$

Independant

(c)
$\text{Det}\left[\left( \begin{array}{ccc} 1 & 1 & 5 \\ 2 & -1 & 1 \\ 1 & -1 & -1 \end{array} \right)\right]=0$ Dependant

Is it ok to use those vector co-efficients in a matrix like that?

 

[rock.freak667]

Where'd you get you those co-efficient matrices from?
What is the definition of vectors being linearly dependent?

 

[Mark44]

That actually works. For example, if you row-reduce the matrix in c), you get
$$\left(\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 0\end{array}\right)$$

This says that c₁ = -2c₃, c₂ = -3c₃, and c₃ = c₃. If you take c₃ = 1, then c₁ = -2 and c₂ = -3. That linear combination of the vectors given in part c results in a sum of 0, thus demonstrating that the set is linearly dependent. Note spelling of "dependent" Gregg. Similar for independent.

 

[Gregg]

That actually works. For example, if you Note spelling of "dependent" Gregg. Similar for independent.

whoops

 

[HallsofIvy]

I think it is always better to use the basic definitions than try to memorize a specific method without understanding it.

The definition of "dependent" for a set of vectors [itex]\left{v_1, v_2, \cdot\cdot\cdot v_n}[/quote] is that there are numbers, $a_1, a_2, \cdot\cdot\cdot a_n$, not all 0, such that \(\displaystyle a_1v_1+ a_2v_2+ \cdot\cdot\cdot a_nv_n= 0[/itex].

For the first problem, { a, 0}, take $a_0= 0$, $a_1= 1$: $a_0a+ a_10= 0(a)+ 1(0)= 0$.

For (b), with a+b, b+c, c+a, if $a_1(a+b)+ a_2(b+ c)+ a_3(c+a)= 0$ then $(a_1+ a_3)a+ (a_1+ a_2)b+ (a_2+ a_3)c= 0$. Since a, b, and c are independent, we must have $a_1+ a_3= 0$, $a_1+ a_2= 0$, and $a_2+ a_3= 0$. Obviously, $a_0= a_1= a_2= 0$ satisfies that but is it the only solution?\)

 

[Mark44]

I couldn't agree with HallsOfIvy more, in what he said about the importance of understanding definitions as opposed to memorizing a technique without understanding why you are doing it. To often students get tangled up in the details of calculating a determinant or row reducing a matrix without understanding what it means that the matrix determinant is zero or why the matrix should be row reduced.

The definition of linear independence of a set of vectors is stated very simply, but there is a subtlety to it that escapes many students. The only thing that distinguishes a set of linearly independent vectors from a set that is linearly dependent is whether the equation $c_1 v_1 + c_2 v_2 + c_3 v_3 + ... + c_n v_n + = 0$ has only one solution (independent vectors) or an infinite number of solutions (dependent vectors).