3) Suppose U is a subspace of V with U + V. Suppose Se L(U,W) and S # 0 (so S is not the zero element of L(U, W), which means Su + 0 for some u e U,where 0 is the zero vector of W). Define T:U - W by (Sv if v e U lo ifveV and v E U. Tv = Prove that T is not a linear map on V.

Suppose V is a finite dimensional vector space and W₁, W₂ are subspaces of V. Consider the map I: W₁ × W₂ → W₁ + W₂ defined by I'(W₁, W₂) = W₁+w₂. Prove the map is linear, and using I show that dim(W₁ + W₂) = dim(W₁) + dim(W₂) – dim(W₁ W₂)

5. Let W1 and W2 be two two-dimensional subspaces of the linear space R°, that is dim W1 W = W1n W2 is a subspace of the dimension at least 1, that is dim W > 1. = dim W2 2. Prove that the intersection of these two-dimensional spaces

2. Let U and W be a subspaces of an n-dimensional vector space V and suppose further that UNW= {0} and U +W =V. Prove that dim(U)+dim(W)=n.