Let V be the two-dimensional subspace of R4 spanned by (0, 1, 0, 1) and (0, 2, 0, 0). Write the vector u = (1, 1, 1, 1) in the form u = v + w, where v is in V and w is orthogonal to every vector in V.
Let V be the two-dimensional subspace of R4 spanned by (0, 1, 0, 1) and (0, 2, 0, 0). Write the
To find v, we need to find the projection of u onto the subspace V, which can be found using the formula:
proj_v(u) = (u . v1 / ||v1||^2) * v1 + (u . v2 / ||v2||^2) * v2
where v1 and v2 are the basis vectors of V, ||v1|| and ||v2|| are their lengths, and the dot (.) denotes the dot product.
Step 1: Find the basis vectors' length
||v1|| = sqrt(0^2 + 1^2 + 0^2 + 1^2) = sqrt(2)
||v2|| = sqrt(0^2 + 2^2 + 0^2 + 0^2) = 2
Step 2: Find the dot products
u . v1 = 01 + 11 + 01 + 11 = 2
u . v2 = 02 + 12 + 00 + 10 = 2
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
