Consider the plane 2 x1-3 x2+4 x3=0 with basis consisting...

Consider the plane $2 x_{1}-3 x_{2}+4 x_{3}=0$ with basis $$\$$ consisting of vectors $\left[\begin{array}{r}8 \\ 4 \\ -1\end{array}\right]$ and $\left[\begin{array}{r}5 \\ 2 \\ -1\end{array}\right] .$ If $[\vec{x}]_{8}=$ $\left[\begin{array}{r}2 \\ -1\end{array}\right],$ find $\vec{x}$

Consider the plane $2 x_{1}-3 x_{2}+4 x_{3}=0$ with basis $$\$$ consisting of vectors $\left[\begin{array}{r}8 \\ 4 \\ -1\end{array}\right]$ and $\left[\begin{array}{r}5 \\ 2 \\ -1\end{array}\right] .$ If $[\vec{x}]_{8}=$
$\left[\begin{array}{r}2 \\ -1\end{array}\right],$ find $\vec{x}$

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Key Concepts

Basis and Coordinate Representation

In linear algebra, a basis for a vector space or a subspace is a set of linearly independent vectors that spans that space. Any vector in the space can be uniquely expressed as a linear combination of the basis vectors with a unique set of coordinates. Understanding this concept allows one to construct a vector from its coordinate representation relative to the chosen basis.

Linear Combination

A linear combination involves multiplying basis vectors by scalars and summing the results to form another vector. This operation is fundamental to expressing vectors in a vector space via a given basis, where the vector is reconstructed as the weighted sum of the basis elements.

Subspaces and Planes

A plane in R³ is an example of a two-dimensional subspace, typically defined by a linear equation. Recognizing that the given plane is a subspace, and that the basis vectors span this plane, is crucial when expressing any vector in the plane as a linear combination of the basis vectors.

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