Show that the vectors v1=(1,1,1), v2=(1,2,3), and...

Show that the vectors $v_{1}=(1,1,1), v_{2}=(1,2,3)$, and $v_{3}=(2,-1,1)$ are linearly independent in $\mathbb{R}^{3}$. Write $v=(1,-2,5)$ as a linear combination of $v_{1}, v_{2}$, and $v_{3}$.

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Key Concepts

Linear Independence

This concept refers to a set of vectors where none of the vectors can be expressed as a linear combination of the others. In the context of vector spaces, demonstrating that a set of vectors is linearly independent ensures that they form a minimal spanning set (or basis) for the subspace they generate.

Determinant

The determinant of a square matrix provides a scalar measure that can reveal whether the set of column (or row) vectors is linearly independent. In particular, in spaces like ?³, if the determinant of the matrix formed by placing the vectors as columns is nonzero, the vectors are linearly independent.

Linear Combination

A linear combination involves expressing a vector as the sum of scalar multiples of other vectors. This concept is central to representing vectors in terms of a chosen set of basis or spanning vectors, and it is used to transition between different coordinate representations within a vector space.

Systems of Linear Equations

In order to write a vector as a linear combination of other vectors, one often sets up a system of linear equations by equating the components of the target vector to the corresponding linear combinations of the basis vectors’ components. Solving these equations yields the coefficients needed for the representation.

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