# Prove Linear Dependence of x^2 + x + 2, x^2 -3x + 1 & 5x^2 -7x + 7

• Naeem
In summary, linear dependence is a mathematical concept where one vector or set of vectors can be expressed as a linear combination of another vector or set of vectors. To prove linear dependence, one can use the concept of linear combinations and find coefficients that result in a linear combination equal to zero. A linear combination is the sum of scalar multiples of vectors. Vectors are considered linearly dependent if there exists at least one non-zero solution to the linear combination equaling zero. To prove linear dependence, a system of equations can be set up using the coefficients of each vector.
Naeem
Q. { x^2 + x + 2 , x^2 -3x + 1, 5x^2 -7x + 7 }

Prove wether or not the above function's are linearly dependent.

Any help shall be very helpful!

You must have had a thought on this problem already -- surely you know, say, the definition, or a relevant theorem?

My thought is to compute the wronskian:

and see if it is equal = 0 it is dependent, if not independent.

That would be an excellent way to do it, you would do well to recall the 3x3 determinant.

## What is linear dependence?

Linear dependence is a mathematical concept that describes how one vector or set of vectors can be expressed as a linear combination of another vector or set of vectors.

## How do you prove linear dependence?

To prove linear dependence, you can use the concept of linear combinations. If you can find coefficients that when multiplied by each vector in the set, result in a linear combination equal to zero, then the vectors are considered linearly dependent.

## What is a linear combination?

A linear combination is the sum of scalar multiples of vectors. For example, in the set of vectors {v1, v2, v3}, a linear combination would be c1v1 + c2v2 + c3v3, where c1, c2, and c3 are scalar coefficients.

## How do you know when vectors are linearly dependent?

Vectors are considered linearly dependent if there exists at least one non-zero solution to the linear combination equaling zero. In other words, if there are coefficients that can be multiplied by the vectors to result in a sum of zero, then the vectors are linearly dependent.

## How can you prove linear dependence of x^2 + x + 2, x^2 -3x + 1, and 5x^2 -7x + 7?

To prove linear dependence of these vectors, we can set up a system of equations using the coefficients of each vector. In this case, we have c1(x^2 + x + 2) + c2(x^2 - 3x + 1) + c3(5x^2 - 7x + 7) = 0. By solving for c1, c2, and c3, we can show that there exists a non-zero solution, thus proving linear dependence.

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