Draw the standard coordinate axes on the same diagram as the...

Draw the standard coordinate axes on the same diagram as the axes relative to u and v. Use these to find $\mathbf{w}$ as a linear combination of u and v. $$\mathbf{u}=\left[\begin{array}{r} 1 \\ -1 \end{array}\right], \mathbf{v}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], \mathbf{w}=\left[\begin{array}{l} 2 \\ 6 \end{array}\right]$$

Draw the standard coordinate axes on the same diagram as the axes relative to u and v. Use these to find $\mathbf{w}$ as a linear combination of u and v.
$$\mathbf{u}=\left[\begin{array}{r}
1 \\
-1
\end{array}\right], \mathbf{v}=\left[\begin{array}{l}
1 \\
1
\end{array}\right], \mathbf{w}=\left[\begin{array}{l}
2 \\
6
\end{array}\right]$$

Key Concepts

Diagram Representation

Diagram representations are graphical methods used to visualize vectors and coordinate systems. Drawing both the standard coordinate axes and axes relative to a given basis (like u and v) aids in understanding how vectors relate to different coordinate systems and in determining the components of a vector with respect to various bases.

Linear Combination

A linear combination involves summing scalar multiples of vectors to form another vector. Expressing a vector as a linear combination of basis vectors is fundamental to understanding coordinate transformations and representations.

Coordinate Axes

Coordinate axes define a frame of reference for vector representation in a space. The standard coordinate axes, usually represented as the x and y axes, provide a familiar system for plotting vectors, while alternative axes (such as those aligned with u and v) help illustrate different bases or transformations.

Vector

A vector is an element of a space that has both magnitude and direction. In two dimensions, vectors are typically represented as ordered pairs and can be added together or scaled by real numbers, forming the basis of many geometric and algebraic operations.

Basis

A basis is a set of linearly independent vectors that span a vector space, meaning that every vector in the space can be uniquely expressed as a linear combination of the basis vectors. In this context, u and v form a basis for the 2-dimensional space.

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