Final answer:
The question asks about decomposing a vector into components within a subspace and perpendicular to it, resulting in two orthogonal components that, when added together, recreate the original vector.
Step-by-step explanation:
Decomposing Vectors
The student's question pertains to the decomposition of vectors with respect to a subspace into orthogonal components. This requires splitting a vector v into two parts, one that lies in the subspace W and the other that is perpendicular to W, denoted as W⊥. The vector v can be decomposed as v = w + z, where w ∈ W and z ∈ W⊥. For example, when decomposing a displacement vector in a plane, one would resolve it into components along the x- and y-axes using the projection of the vector onto these axes. To find vector w, the orthogonal projection of v onto W is calculated, whereas vector z is found by subtracting w from v. The result gives two orthogonal vectors that sum up to the original vector v.
Example
To calculate the decomposition of vector A with respect to Line C, first determine the direction of C and project A onto this direction to find w. Next, calculate z as the difference between A and w, which will be perpendicular to Line C. By vector addition, these components yield the original vector A.