Linear dependence of non-numerical objects

[ashina14]

Homework Statement

Suppose V = 2^Ω where Ω = {red,blue,yellow,green}. Verify whether u={blue,green}, v={red,yellow,green}, and w={blue,yellow} are linearly independent in V

The Attempt at a Solution

I let
red = a1
blue = a2
yellow =a3
green = a4

therefore Ω = {a1, a2, a3, a4}

so now u={a2,a4}, v={a1,a3,a4}, w={a2,a3}

These vectors can also be written as
u=0.a1+1.a2+0.a3+1.a4
v=1.a1+0.a2+1.a3+1.a4
w=0.a1+1.a2+1.a3+0.a4

writing these vectors out in one matrix gives us
(0 | 1 | 0
1 | 0 | 1
0 | 1 | 1
1 | 1 | 0) if we're supposed to write the vectorsas columns

then i reduced it to REF to get
(1 | 0 | 0
0 | 1 | 0
0 | 0 | 1
0 | 0 | 0)

But then what do I do? How do I know what is dependent or not? I thought I could just say a1=0, a2=0,a3=0,a4=free. That shows these are independent but there is an empty row surely something's dependent??

 

[mighty2000]

Thank you for your question. To determine whether the vectors u, v, and w are linearly independent in V, we can use the definition of linear independence.

Linear independence means that no vector in a set can be written as a linear combination of the other vectors in the set. In other words, there is no non-trivial solution to the equation a1u + a2v + a3w = 0, where a1, a2, and a3 are scalars.

In your attempt, you have correctly written the vectors u, v, and w as columns of a matrix and reduced it to REF form. However, the empty row does not necessarily mean that the vectors are dependent. It simply means that there are no more equations that can be used to reduce the matrix further.

To determine whether a set of vectors is linearly independent, we can also use the determinant. If the determinant of the matrix formed by the vectors is non-zero, then the vectors are linearly independent. In your case, the determinant is equal to 1, which means that the vectors u, v, and w are linearly independent in V.

I hope this helps to clarify your understanding. Let me know if you have any further questions.