Final answer:
The student is asked to find an explicit description of the null space of a matrix by listing vectors that span it. They would then apply matrix row reductions to find a basis for the null space. The null vector, null space, and operations like dot and cross products are all essential concepts related to vectors in the Cartesian coordinate system.
Step-by-step explanation:
The question is asking to provide an explicit description of the null space (often denoted as nul(A)) of a matrix A by listing vectors that span this null space. The null space consists of all vector solutions to the homogeneous equation A\(\mathbf{v}\) = \(\mathbf{0}\), where A is a matrix and \(\mathbf{v}\) is a vector in the null space. To find a basis for the null space, you would typically perform row reductions on the matrix A to bring it into reduced row echelon form and then solve for the vectors that satisfy the equation A\(\mathbf{v}\) = \(\mathbf{0}\).
The null vector, as mentioned in the reference material, differs from the null space in that the null vector is simply a vector with all zero components, serving as the identity element for vector addition. This is different from the null space of a matrix, which may consist of non-zero vectors that still result in a zero vector when multiplied with the matrix.
Regarding vectors and their operations, finding angles with the axes, orthogonal vectors, and vector products all relate to understanding the properties of vectors within the context of a Cartesian coordinate system. Orthogonality, for example, is determined by the dot product being zero, while cross products give a vector that is orthogonal to both original vectors.