Writing vectors as a linear combination involves expressing a vector as a sum of scalar multiples of other vectors, typically basis vectors.

Let's break this down with a general example. Suppose you have a vector **v** in a 2-dimensional space and you want to express it as a linear combination of two basis vectors **b₁** and **b₂**. The general form is:

\[
\mathbf{v} = c_1\mathbf{b}_1 + c_2\mathbf{b}_2
\]

Where:
- \(\mathbf{v}\) is the vector you want to express.
- \(c_1\) and \(c_2\) are scalar coefficients.
- \(\mathbf{b}_1\) and \(\mathbf{b}_2\) are the basis vectors.

### Example 1: Expressing a 2D Vector

Let’s say \(\mathbf{v} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}\), and we choose the standard basis vectors \(\mathbf{b}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\mathbf{b}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\).

We can write:

\[
\mathbf{v} = 4\mathbf{b}_1 + 3\mathbf{b}_2 = 4\begin{pmatrix} 1 \\ 0 \end{pmatrix} + 3\begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 4 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 3 \end{pmatrix} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}
\]

So, \(\mathbf{v}\) is written as a linear combination of \(\mathbf{b}_1\) and \(\mathbf{b}_2\).

### Example 2: Expressing a 3D Vector

For a 3D vector \(\mathbf{v} = \begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix}\) and the standard basis vectors \(\mathbf{b}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\), \(\mathbf{b}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\), and \(\mathbf{b}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\), the linear combination is:

\[
\mathbf{v} = 2\mathbf{b}_1 - 1\mathbf{b}_2 + 5\mathbf{b}_3
\]

This represents \(\mathbf{v}\) as:

\[
\mathbf{v} = 2\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} - 1\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + 5\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ -1 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ 5 \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix}
\]

### Summary
To express any vector as a linear combination:
1. Identify the basis vectors.
2. Determine the scalar coefficients.
3. Combine them accordingly.

Would you like more details on a specific example or a different type of vector space?

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### Questions for Further Understanding:
1. How can you determine if a set of vectors forms a basis for a vector space?
2. What is the process for finding the coefficients \(c_1\) and \(c_2\) when the basis vectors are not standard?
3. Can any vector be written as a linear combination of non-basis vectors?
4. What is the significance of a linear combination in terms of vector spaces?
5. How do you express a vector in a higher-dimensional space as a linear combination?
6. What does it mean for a set of vectors to be linearly dependent?
7. How can you apply linear combinations in solving systems of linear equations?
8. What are some practical applications of expressing vectors as linear combinations?

### Tip:
Remember, if a vector can be written as a linear combination of a set of vectors, those vectors span the vector's space. This concept is crucial in understanding the structure of vector spaces.

Learn how to express vectors as linear combinations with detailed examples. Understand the concept of basis vectors, scalar coefficients, and their role in vector spaces. Suitable for advanced high school students and above.

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