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

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

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Key Concepts

Vector Representation

Representing vectors involves plotting them as directed line segments with magnitude and direction. When using non-standard bases, each vector's coordinates indicate how far and in which direction to move along each of the new axes, enabling one to visualize and locate any vector within that coordinate system.

Coordinate Transformation

This concept refers to the redefinition or change of the coordinate system. When standard axes are replaced by non-standard basis vectors like u and v, the representation and plotting of vectors such as w must be interpreted through their components relative to these new axes.

Linear Combination

A linear combination involves scaling vectors and adding them together. In this context, the vector w is obtained by multiplying the basis vectors by appropriate scalars and summing the results, illustrating how any vector can be constructed from the basis vectors.

Basis Vectors

Basis vectors form the building blocks of a vector space; any vector in the space can be uniquely expressed as a linear combination of them. When redefining coordinate axes using different vectors (such as u and v), these vectors serve as the new 'x' and 'y' axes, giving a different perspective on location and direction within the space.

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