Addition of Two Vectors

Given two vectors $X = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $Y = \begin{bmatrix} -4 \\ 6 \end{bmatrix}$, compute $X + Y$.

This problem involves simple vector addition, a fundamental operation in vector mathematics and linear algebra. Adding vectors involves adding corresponding components from each vector to form a new vector. For two-dimensional vectors like the ones provided, the addition is straightforward: simply add the first component of the first vector to the first component of the second vector to get the first component of the result. Then, do the same for the second components. This process illustrates the principle of superposition, which is crucial in various applications of linear algebra.

Understanding vector addition is vital as it forms the basis for more complex operations, such as vector transformations, linear mappings, and operations involving matrices. This problem also helps solidify comprehension of how vectors can combine within a vector space, influencing concepts like span and linear combinations. These are key concepts for solving problems in physics, engineering, and computer science where multi-dimensional analysis is necessary.

Additionally, mastering vector operations such as addition can make it easier to approach more advanced topics in linear algebra, such as vector projections, subspaces, and understanding the geometry of linear transformations. This foundational knowledge supports learning about vector spaces' properties, paving the way to tackle advanced mathematical problems more effectively.