Matrix Algebra; Linear Dependece; 3 Vectors; In R^4

[bmed90]

Homework Statement

You have 3 Vectors say v1,v2,v3 in R^4 meaning they each have 4 components. How do you determine if they are linearly dependent or independent? And usually how do you denote 4 components? I know everyone knows how to denote three components which is just x,y,z but idk what to call the 4th one. The question is simple, can someone just give me a brief play by play on how to determine the dependence or independence? My book is not very much help and I even googled it.

Homework Equations

The Attempt at a Solution

 

[NewtonianAlch]

You can just call them x1, x2, x3, x4 - in fact you can call them anything you want to, but it's best to stick to standard conventions so it doesn't confuse anyone.

So it would be in a column: <x1,x2,x3,x4>

You need to create a matrix with the 3 vectors (3 columns and 4 rows because of 4 components), and perform row reduction on them. This will help you determine if they are linearly dependent or not.

 

[HallsofIvy]

Are you saying that your book does not give the definition of "independence" and "dependence"? I find that had to believe! And it is all you need.
A set of vectors, $\{v_1, v_2, ..., v_n\}$, is "dependent" if and only if there exist a set of numbers, $\{a_1, a_2,..., a_n\}$, not all 0, such that $a_1v_1+ a_2v_2+ ...+ a_nv_n= 0$.

If you have vectors in $R^4$, each vector has four components and so, with three vectors, you would have three equations in four unknowns. It should be easy to find many values of the unknowns that satify those equations.

("not all 0" added after Mark44 pointed out my error. Thanks.)

 

[Mark44]

A set of vectors, $\{v_1, v_2, ..., v_n\}$, is "dependent" if and only if there exist a set of numbers, $\{a_1, a_2,..., a_n\}$, such that $a_1v_1+ a_2v_2+ ...+ a_nv_n= 0$.

There must be a typo here. Using this definition, I would conclude wrongly that x = <1, 0> and y = <0, 1> are linearly dependent vectors in R². Clearly, the numbers a = 0 and b = 0 are a solution for the equation ax + by = 0. In fact, there are no other solutions for the constants a and b in this equation.

If the equation $a_1v_1+ a_2v_2+ ...+ a_nv_n= 0$ has only a single solution (the trivial solution, a₁ = 0 = a₂ = ... = a_n) for the constants a₁, ..., a_n, then the vectors $\{v_1, v_2, ..., v_n\}$ are linearly independent.

If this equation has more than one solution for the constants, a₁, ..., a_n, then this set of vectors is linearly dependent.

 

[vela]

If you have vectors in $R^4$, each vector has four components and so, with three vectors, you would have three equations in four unknowns. It should be easy to find many values of the unknowns that satify those equations.

You're thinking of the case of four vectors in R³. With three vectors in R⁴, you get four equations with three unknowns.