Prove whether or not the following are linearly independent

Thread starter skysurani
Start date Apr 14, 2005
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Independent Linearly

In summary, linear independence refers to a set of vectors that cannot be written as a linear combination of each other. To prove if vectors are linearly independent, one can use the linear independence test. A set of 2 or 3 vectors can be linearly independent or dependent. Linear independence is important in linear equations as it determines the number of solutions. Linearly independent vectors cannot be in the same direction.

Apr 14, 2005

skysurani

f1(x)= (sqrtx) + 5,
f2(x)= (sqrtx) + 5x
f3(x)= x-1

i don't know how to start

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Apr 14, 2005

Hurkyl

Staff Emeritus

Science Advisor

Gold Member

Applying definitions is often a good place to start.

Apr 14, 2005

graphic7

Gold Member

First of all, you haven't stated which of these functions are in the set(s) that you are checking for linear independence.

1. What does it mean for vectors to be linearly independent?

Linear independence refers to a set of vectors that cannot be written as a linear combination of each other. This means that none of the vectors can be created by multiplying another vector by a scalar and adding it to the other vectors in the set.

2. How do you prove whether or not vectors are linearly independent?

To prove if vectors are linearly independent, you can use the linear independence test. This involves setting up an equation with the vectors as coefficients and solving for a combination of scalars. If the only solution is when all the scalars are equal to 0, then the vectors are linearly independent.

3. Can a set of 2 or 3 vectors be linearly independent?

Yes, a set of 2 or 3 vectors can be linearly independent. The number of vectors in a set does not determine their linear independence. It is possible for a set of 2 or 3 vectors to be linearly dependent or independent.

4. How does linear independence relate to linear equations?

Linear independence is important in linear equations because it determines whether a system of equations has a unique solution. If the vectors in a system are linearly dependent, there are infinite solutions. If the vectors are linearly independent, there is only one solution.

5. Can linearly independent vectors be in the same direction?

No, linearly independent vectors cannot be in the same direction. If two vectors are in the same direction, one can be represented as a scalar multiple of the other, making the set linearly dependent. Linearly independent vectors must have different directions.