# Linear independance

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?

Last edited:

rock.freak667
Homework Helper
Where'd you get you those co-efficient matrices from?
What is the definition of vectors being linearly dependent?

Mark44
Mentor
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.

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

whoops

HallsofIvy
The definition of "dependent" for a set of vectors $\left{v_1, v_2, \cdot\cdot\cdot v_n}[/quote] is that there are numbers, [itex]a_1, a_2, \cdot\cdot\cdot a_n$, not all 0, such that $$\displaystyle a_1v_1+ a_2v_2+ \cdot\cdot\cdot a_nv_n= 0[/itex]. For the first problem, { a, 0}, take $a_0= 0$, $a_1= 1$: $a_0a+ a_10= 0(a)+ 1(0)= 0$. For (b), with a+b, b+c, c+a, if $a_1(a+b)+ a_2(b+ c)+ a_3(c+a)= 0$ then $(a_1+ a_3)a+ (a_1+ a_2)b+ (a_2+ a_3)c= 0$. Since a, b, and c are independent, we must have $a_1+ a_3= 0$, $a_1+ a_2= 0$, and $a_2+ a_3= 0$. Obviously, $a_0= a_1= a_2= 0$ satisfies that but is it the only solution?$$
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).