Linear Independence of Vectors

Are the vectors $\mathbf{v}_1 = (1, 2, 3)$, $\mathbf{v}_2 = (2, -1, 4)$, and $\mathbf{v}_3 = (0, 5, 2)$ linearly independent?

To determine whether the vectors v1, v2, and v3 are linearly independent, we employ a fundamental concept from linear algebra. Linear independence of vectors essentially means that no vector in the set can be written as a linear combination of the others. If we can express any vector as a combination of the others, the vectors are linearly dependent.

A common approach to testing linear independence is to arrange the vectors as columns in a matrix and then compute the determinant. If the determinant is non-zero, the vectors are linearly independent. Another method is to perform row reduction on the matrix to achieve echelon form and assess the rank. If the matrix has full rank, meaning the rank equals the number of vectors, they are independent.

Understanding linear independence is crucial as it underlies many applications in linear algebra, including determining the span of a set of vectors, identifying basis vectors in vector spaces, and solving systems of linear equations. Grasping this concept will enhance your problem-solving skills in matrix theory and beyond.