Understanding Notation and Proving Linear Independence

In summary, the conversation discusses the concept of linear independence and how removing a vector from a set can still maintain linear independence, as shown in the example of {a,b,c,d} and its subsets. The conversation also suggests using a proof by contradiction to prove this concept.
  • #1
cateater2000
35
0
Hi I just need some help on understanding some general notation in this quesiton:

Prove if {x_1,x_2,..,x_m} is linearly independent then so is {x_1,x_2,...,x_i-1, x_i+1,...,x_m} for every i in {1,2,...,m}.


I don't really understand what the difference between {x_1,x_2,...,x_i-1, x_i+1,...,x_m} for every i in {1,2,...,m} and {x_1,x_2,..,x_m} is.

Any help clarifying this would be great, and any hints for the question would be must appreciated, thanks.
 
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  • #2
The second set omits the i'th vector.

Eg the large set is {a,b,c,d} and there are 4 other sets: {b,c,d}, {a,c,d}, {a,b,d}, {a,b,c}
 
  • #3
If you understand that, you can easily prove this by contradiction.
Suppose one of the smaller sets is linearly dependent, then one of its elements is a lineair combination of the others. What does that tell you about the larger set then?
 

FAQ: Understanding Notation and Proving Linear Independence

What is a linear independence proof?

A linear independence proof is a mathematical demonstration that shows a set of vectors in a vector space is linearly independent, meaning that no vector in the set can be written as a linear combination of the other vectors in the set.

Why is it important to prove linear independence?

Proving linear independence is important because it allows us to determine if a set of vectors forms a basis for a vector space, which is crucial in various areas of mathematics and science, such as linear algebra, differential equations, and physics.

What is the process for proving linear independence?

The process for proving linear independence involves setting up a linear combination equation with the vectors in the set and solving for the coefficients. If the only solution is when all coefficients are equal to zero, then the set of vectors is linearly independent.

Can you give an example of a linear independence proof?

For example, if we have the vectors (1,0,0), (0,1,0), and (0,0,1), we can set up the equation c1(1,0,0) + c2(0,1,0) + c3(0,0,1) = (0,0,0). Solving for c1, c2, and c3, we get c1 = c2 = c3 = 0, showing that the vectors are linearly independent.

What are some applications of linear independence proofs?

Linear independence proofs have various applications in fields such as physics, engineering, computer science, and economics. For example, in physics, linear independence is used to determine the equations of motion for a system, while in economics, it is used to analyze the relationships between different variables.

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