STEM Lessons for College Students

Linear Algebra Example Problems – Linearly Independent Vectors #2

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This example also works with a set of vectors, however, one of these vectors contains the variable “alpha” as one of its elements.

We find the value of alpha that makes the set of vectors linearly dependent. A set of vectors is linearly dependent when their linear combination is equal to an all-zero vector without all combining coefficients being zero.

So, to determine when these vectors are dependent, we just have to construct and solve a homogeneous system of equations. We row reduce the corresponding augmented matrix and find the value of alpha that results in a free variable. With a free variable in the solution, we have an infinite family of solutions, not just the trivial all-zero vector, resulting in a set of vectors that is linearly dependent.

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