Determine Scalars for Vector Equation

Given vectors $V_1 = \begin{bmatrix} 1 \\ -2 \\ -5 \end{bmatrix}$, $V_2 = \begin{bmatrix} 2 \\ 5 \\ 6 \end{bmatrix}$, and $B = \begin{bmatrix} 7 \\ 4 \\ -3 \end{bmatrix}$, determine if there exist scalars $x_1$ and $x_2$ such that $x_1 V_1 + x_2 V_2 = B$.

This problem involves determining if a target vector can be expressed as a linear combination of other given vectors. At its core, this is a classic problem in linear algebra which tests understanding of linear combinations and spans. A vector can be written as a linear combination of other vectors if and only if it is within the span of those vectors. This involves solving the equation of the form x1 times vector1 plus x2 times vector2 equals the target vector, and checking if a solution exists.

The challenge is to set this up correctly. One can approach this problem by equating components and setting up a system of linear equations, then solving for the scalars using methods such as substitution or elimination. An alternative method is to augment these vectors into a matrix form and apply techniques from linear algebra to solve the resultant system. Understanding these techniques is crucial as they apply broadly in many areas, including solving larger and more complex systems of equations.

The concepts derived from this problem, such as spans, basis, and the geometrical interpretations of vector spaces, are foundational in linear algebra. They are not only important theoretically but are also widely applicable in computer graphics, data science, and systems modelling. Thus, gaining a firm grasp by practicing these problems enhances one's ability to address various practical and more advanced theoretical problems later on.