Calculating Linear Combination of Vectors

Calculate the linear combination $3X + 2Y$ for the vectors $X = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $Y = \begin{bmatrix} -4 \\ 6 \end{bmatrix}$.

In linear algebra, linear combinations are fundamental concepts used to express one vector as a combination of two or more vectors, scaled by respective coefficients. Understanding linear combinations is essential for vector space analysis, as they are used in defining concepts like span, basis, and vector representation in vector spaces. This problem helps reinforce the idea that vectors can be manipulated using basic arithmetic operations like addition and scalar multiplication. Calculating a linear combination involves multiplying each vector by its respective scalar and adding the resulting products together. In this case, the vectors X and Y are multiplied by scalars 3 and 2, respectively. By working through the process of combining these vectors, you solidify your understanding of how different scalars and vectors interact within a vector space.

Thinking in terms of geometric interpretation, each vector can be visualized as an arrow in two-dimensional space. The linear combination forms a new vector through scaling these arrows in size and direction, then summing them up. This geometric view helps with intuition as you study more complex vector operations, making it easier to understand transformations and vector spaces in higher dimensions. This exercise is not just about calculation, but about visualizing and understanding vector relations, which is pivotal as you move through topics in linear algebra.