91Ó°ÊÓ

When dealing with coordinate vectors in linear algebra, the term "basis vectors" comes up quite often. A set of vectors that span a vector space forms a basis. In other words, any vector in the vector space can be expressed as a combination of these basis vectors.
For the exercise given, our basis \( \mathcal{B} \) consists of three specific vectors. These vectors are \( \begin{pmatrix} -1 \ 2 \ 0 \end{pmatrix} \), \( \begin{pmatrix} 3 \ -5 \ 2 \end{pmatrix} \), and \( \begin{pmatrix} 4 \ -7 \ 3 \end{pmatrix} \). They are used to represent other vectors in the space.

• Each of these vectors is called a basis vector.
• They are linearly independent, meaning no vector can be written as a combination of the others.
• They are used to define a unique coordinate system in the space.

Understanding these vectors allows us to express any vector in that space as a linear combination of these three vectors.

The idea of a linear combination is essential in understanding how vectors relate to each other in space. A linear combination of vectors involves adding together multiples of those vectors.
For example, consider the vectors \( \mathbf{b}_1, \mathbf{b}_2, \) and \( \mathbf{b}_3 \) and constants \( c_1, c_2, \) and \( c_3 \) as scalars. A vector \( \mathbf{x} \) is a linear combination of these vectors if:\[ \mathbf{x} = c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2 + c_3 \mathbf{b}_3\]

• This notation tells us how to construct \( \mathbf{x} \) using scales of the other vectors.
• The coefficients \( c_1, c_2, \) and \( c_3 \) can be found in the coordinate vector \([\mathbf{x}]_{\mathcal{B}}\).
• The resulting vector is a new point in the vector space based on these coefficients.

In this exercise, by calculating each vector's contribution through scalar multiplication, the linear combination produces the final vector \( \mathbf{x} \) as expressed in the standard basis.

The standard basis in a vector space is an easily recognizable set of vectors. In \( \mathbb{R}^3 \), the standard basis consists of vectors that point along each of the coordinate axes: \( \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}, \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}, \) and \( \begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix} \).
These basis vectors are used in many problems because they simplify complex equations into more manageable forms. They are the default coordinate system in three-dimensional space.

• They make it easy to visualize vector spaces.
• Vectors expressed in terms of the standard basis are often easier to manipulate and understand.
• Conversion from a custom basis to the standard basis is a common task, like in our exercise.

By converting each of the custom basis vectors into their linear combination of the standard basis, we simplify our vector \( \mathbf{x} \) into something easier to interpret and use in equations or geometric interpretations.