Use a sheet of graph paper and the technique presented in...

Use a sheet of graph paper and the technique presented in Figure 7 to find a linear combination of $\mathbf{x}$ and $\mathbf{y}$ that equals $\mathbf{z}$, if possible. Verify your result analytically. $\mathbf{x}=\left(\begin{array}{l}1 \\ 3\end{array}\right), \mathbf{y}=\left(\begin{array}{l}4 \\ 1\end{array}\right), \mathbf{z}=\left(\begin{array}{r}6 \\ -4\end{array}\right)$

Use a sheet of graph paper and the technique presented in Figure 7 to find a linear combination of $\mathbf{x}$ and $\mathbf{y}$ that equals $\mathbf{z}$, if possible. Verify your result analytically.
$\mathbf{x}=\left(\begin{array}{l}1 \\ 3\end{array}\right), \mathbf{y}=\left(\begin{array}{l}4 \\ 1\end{array}\right), \mathbf{z}=\left(\begin{array}{r}6 \\ -4\end{array}\right)$

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Key Concepts

Analytic Verification

Analytic verification involves setting up and solving a system of linear equations that arise from equating the components of the linear combination to the components of the target vector. This method confirms whether the computed coefficients correctly express the target vector as a linear combination of the given vectors.

Graphical Representation in Vector Addition

Graphical techniques aid in visualizing vector operations such as addition and scalar multiplication. Drawing vectors on graph paper helps to intuitively find a linear combination by showing how scaled vectors add to form a target vector.

Linear Combination

A linear combination involves creating a new vector by multiplying given vectors by scalars and then adding the results together. This concept is fundamental in linear algebra as it describes how any vector in a vector space can potentially be expressed in terms of a specific set of vectors.

Span

The span of a set of vectors is defined as the collection of all possible vectors that can be formed as linear combinations of these vectors. Determining the span of vectors is essential for understanding concepts like basis and dimension in vector spaces.

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