91Ó°ÊÓ

In linear algebra, basis vectors are a set of vectors that are both linearly independent and span the vector space. These bases form a foundation upon which any vector in that space can be constructed. In our example, \( \mathbf{b}_1, \mathbf{b}_2, \mathbf{b}_3 \) are the basis vectors for the vector space. Any vector in this vector space can be expressed as a linear combination of these basis vectors.

• Spanning: The basis vectors can reach any point in the vector space through their linear combinations.
• Independence: No vector in the set can be written as a combination of the others.

Understanding basis vectors is crucial as they allow us to simplify complex vector spaces, making calculations easier and more intuitive.

A linear combination involves taking several vectors and combining them by multiplying each vector by a coefficient and then summing the results. In a linear combination, we form new vectors in the same space by scaling existing vectors and adding them together.
For the vector \( \mathbf{x} \), as indicated in the exercise, we expressed it using coefficients \( c_1, c_2, \) and \( c_3 \) with respect to the basis vectors:\[ \mathbf{x} = c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2 + c_3 \mathbf{b}_3 \]Finding the right linear combination means determining the correct coefficients, a central problem in linear algebra that often requires solutions to systems of equations.

Gaussian Elimination is a method used in linear algebra to solve systems of linear equations. It's a step-by-step procedure for transforming a matrix into a form where the solution can be easily spotted. This matrix form is called the row echelon form or reduced row echelon form.
In this exercise, we use Gaussian elimination to solve for the coefficients \( c_1, c_2, \) and \( c_3 \). The process involves:

• Arranging the equations into a matrix format (known as an augmented matrix).
• Performing row operations to simplify the matrix, ideally to an upper triangular form.
• Using back substitution to solve for each variable.

This method is efficient and widely used in computational algorithms for solving systems of equations.

A coordinate system allows us to assign a unique tuple of numbers to each point in a space in a consistent manner. In our context, it helps represent vectors in terms of a particular basis.
When we found the coordinate vector \( [\mathbf{x}]_{\mathcal{B}} \) of \( \mathbf{x} \) relative to the basis \( \mathcal{B} \), we essentially expressed \( \mathbf{x} \) in the coordinate system defined by the basis vectors.
This is important because it provides us with a way to express vectors in terms of readily understood scalar multiples:\[ [\mathbf{x}]_{\mathcal{B}} = \begin{bmatrix} -1 \ 2 \ 3 \end{bmatrix} \]With this representation, vector spaces become more manageable, allowing for clear analysis and solution of complex problems.