Question

Question 1. Let V be a finite dimensional vector space over a field F and let W be a subspace of Prove that the quotient space V/W is finite dimensional and dimr(V/IV) = dimF(V) _ dimF(W). Hint l. Start with a basis A = {wi, . . . , w,n} for W and extend it to a basis B = {wi , . . . , wm, V1 , . . . , va) for V. Hint 2. Our goal is to prove that C =何+1. . . . , vn + W} is a basis for V/ W. What do we need do in order to show C is a basis? Hint 3. Remember ·i, + W-6 + 11, if and only if vi-V2 E W, ·0+ W is the zero vector in V/W,

Answer 1

V is finite dimensional. Since W is a subspace of V so W is also finite dimensional. Suppose dim_F W = m Then m greaterthanorequalto dim_F V Let dim_F V = m + n Let A = {W^vector_1, ellipsis, W^vector_m} be a basis for W. By Extension theorem, we extend A and let B = {W^vector_1, ellipsis, W^vector_m, V^vector_1, ellipsis, V^vector_n} be a basis for V. Let u^vector + W elementof V/W. Then u^vector elementof V Then we can express u^vector as u^vector = c_1 w^vector_1 + ellipsis + c_m w^vector_m + d_1 v^vector_1 + ellipsis + d-n v^vector_n where c_1, ellipsis, c_m, d_1, ellipsis, d_n elementof F Therefore u^vector - (d_1 v^vector_1 + ellipsis + d_n v^vector_n) = c_1 w^vector_1 + ellipsis + c_m w^vector_m elementof W (since W is a subspace) That is, u^vector - (d_1 v^vector_1 + ellipsis + d_n V^vector) elementof W \r\n Therefore u + W = (d_1 v^vector_1 + ellipsis + d_n V^vector_n) + W [By