Final answer:
The question involves a subspace in four-dimensional space represented by the equation 2x - 3y - z + w = 0. We discussed the subspace, null vector, and three-dimensional Cartesian coordinates as well as their relevance in this context.
Step-by-step explanation:
The student is asking about a subspace specified by a linear equation in a four-dimensional space (R⁴). A subspace in linear algebra is a set of vectors that are closed under vector addition and scalar multiplication. In this case, the subspace N consists of all vectors (x, y, z, w) that satisfy the equation 2x - 3y - z + w = 0. Any vector in this subspace, when added to another vector in the subspace or multiplied by a scalar, will also be in the subspace.
The generalization of the number zero to vector algebra introduces the concept of a null vector. The null vector has all components equal to zero and, hence, no length or direction, symbolized as 0 = 0î + 0ʟ + 0k.
In three-dimensional Cartesian coordinates, locations are determined by the coordinates (x, y, z) and characterized using the standard right-handed coordinate system where the unit vectors are designated as î for the x-axis, ʟ for the y-axis, and k for the z-axis. However, since we're working in four dimensions in this case, there would be an additional unit vector for the fourth dimension, typically denoted as j.