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.
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.
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.
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.
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.