Let V be a vector space over a field F and let W and Y be su

Let V be a vector space over a field F and let W and Y be subspaces of V satisfying W + Y = V. Let Y\' be a complement of Y in V and let Y\" be a complement of W intersect Y in W. Show that Y\' congruent to y\".

Solution

let v be a vector space over a field f and let w and y be subspaces o v satisfying w+y=v

Example II: Let the field be R again, but now let the vector space be the Cartesian plane R². Take W to be the set of points (x,y) of R² such that x = y. Then W is a subspace ofR².

Proof:

In general, any subset of the real coordinate space Rⁿ that is defined by a system of homogeneous linear equations will yield a subspace. (The equation in example I was z = 0, and the equation in example II was x = y.) Geometrically, these subspaces are points, lines, planes, and so on, that pass through the point 0.