91Ó°ÊÓ

In linear algebra, the concept of "Span" plays an essential role in understanding vector spaces. The span of a set of vectors is the collection of all possible linear combinations of those vectors. You can think of it as all the "reachable" vectors you can get by scaling and adding together the vectors in your set.
For example, if you have two vectors, \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \), any vector in their span can be written as \( c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 \), where \( c_1 \) and \( c_2 \) are scalars. In the problem provided, the vectors are \( \begin{bmatrix} 3 \ 1 \ 4 \end{bmatrix} \) and \( \begin{bmatrix} 5 \ 1 \ 6 \end{bmatrix} \).

• To determine if a vector, \( \mathbf{w} \), is in the span of \( \mathcal{B} \), check if you can write \( \mathbf{w} \) as a linear combination of the vectors in \( \mathcal{B} \).
• If you can find scalars \( c_1 \) and \( c_2 \) such that \( c_1 \begin{bmatrix} 3 \ 1 \ 4 \end{bmatrix} + c_2 \begin{bmatrix} 5 \ 1 \ 6 \end{bmatrix} = \begin{bmatrix} 1 \ 3 \ 4 \end{bmatrix} \), then \( \mathbf{w} \) is in \( \text{span}(\mathcal{B}) \).

In this task, we find that \( \mathbf{w} \) is indeed in the span of \( \mathcal{B} \) because such scalars exist.

The coordinate vector in a given basis represents how a vector in a span can be uniquely expressed using the vectors of that basis, similar to how you represent a point on a graph using x and y coordinates.
In simple terms, a coordinate vector \([\mathbf{w}]_{\mathcal{B}}\) tells you how much of each vector in the basis \( \mathcal{B} \) is needed to construct \( \mathbf{w} \).
To find the coordinate vector, you express \( \mathbf{w} \) as a linear combination of the basis vectors, as shown in the solution. For instance, if \( \mathbf{w} = c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2 \), then the coordinate vector is \( \begin{bmatrix} c_1 \ c_2 \end{bmatrix} \).

• In our problem, finding \( c_1 \) and \( c_2 \) gives us the coordinate vector \( [\mathbf{w}]_{\mathcal{B}} = \begin{bmatrix} 7 \ -4 \end{bmatrix} \).
• This means \( \mathbf{w} \) can be constructed by adding \( 7 \) times the first basis vector and \( -4 \) times the second basis vector.

Remember, the coordinate vector provides a compact way of showing how the target vector is built from the basis.

A linear combination involves creating a new vector by multiplying each vector in a set by a scalar and then summing the results. This is a fundamental operation in linear algebra.
To construct a linear combination of vectors, you take each vector in your set and pair it with a real number (a scalar). Then, sum all these products. This process generates a new vector.
For instance, given vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \), their linear combination is \( c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 \). Here:

• \( c_1 \) and \( c_2 \) are scalars that can be any real number.
• The resulting vector depends on the values of \( c_1 \) and \( c_2 \).

In our problem, finding the correct scalars \( c_1 = 7 \) and \( c_2 = -4 \) was crucial. They transformed the problem vector \( \mathbf{w} \) into a linear combination of the basis vectors, confirming it belongs to the span. Such combinations are critical in defining vector spaces and solving systems of linear equations.