Draw the coordinate axes relative to u and v and locate w....

Draw the coordinate axes relative to u and v and locate w.
$$\mathbf{u}=\left[\begin{array}{r}
1 \\
-1
\end{array}\right], \mathbf{v}=\left[\begin{array}{l}
1 \\
1
\end{array}\right], \mathbf{w}=2 \mathbf{u}+3 \mathbf{v}$$

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Key Concepts

Linear Combination

A linear combination involves multiplying vectors by scalars and adding the results together. In the problem, vector w is formed by a linear combination of the basis vectors u and v. This concept is fundamental in expressing vectors relative to a given set of basis vectors.

Coordinate Systems Relative to a Basis

Choosing a non-standard basis such as u and v allows for the construction of customized coordinate axes. This concept underscores how different bases provide different coordinate representations, and it is particularly useful in contexts like geometry and physics where a tailored coordinate system can simplify the problem.

Vector Representation

This concept involves expressing quantities as vectors, which have both magnitude and direction. In the context of linear algebra, vectors are often represented in coordinate systems, and understanding how to denote their components is key to visualizing and manipulating them in geometric space.

Basis

A basis of a vector space is a set of vectors, in this case u and v, that are linearly independent and span the entire space. Using a basis provides a framework for representing any vector in the space as a unique linear combination of the basis vectors, which is critical in understanding coordinate systems.

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