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Math 108A: Fall 2022

Practice Final

1.  (10 points) Determine whether the following statements are true or false.  You don’t have to justify your answers.

(a) If U1 ,U2 ,W are subspaces of a vector space V such that U1 ⊕ W = U2 ⊕ W, then U1  = U2 .

(b) If v1 ,v2 ,v3  ∈ V are linearly independent, then v1 − v2 , v2 − v3 , v3 − v1  are linearly independent as well.

(c) Let V,W be finite dimensional vector spaces over F and T ∈ L(V,W). Then V is isomorphic to (ker T) × (rangeT).

(d) Let V,W be finite dimensional vector spaces and let T ∈ L(V,W) be an isomor- phism. Then T\  ∈ L(W\ ,V\ ) is an isomorphism as well.

(e) The subset {p ∈ P(R)  :  deg(p) is even or −∞} ⊆ P(R) is a subspace of P(R).

2.  (12 points) Let V and W be finite dimensional non-zero vector spaces over F, and let U ⊆ V be a subspace. Define

A := {T ∈ L(V,W)  :  U ⊆ kerT}.

(a)  (4 points) Show that A is a subspace of L(V,W).

(b)  (8 points) If U is a proper subspace of V (i.e., U ⊊ V), prove that A is a non-zero subspace.

3.  (8 points) Let V be a vector space over F. Prove that V and L(F,V) are isomorphic. Note: You may not assume that V is finite dimensional.

4.  (10 points) Let V be a finite dimensional vector space and let φ ∈ V\  be non-zero. Prove that dim[V/(kerφ)] = 1.

5.  (12 points) Let V be a finite dimensional vector space over F, and let U ⊆ V be a subspace. Prove that U0  = {0} if and only if U = V .

6.   (a)  (2 points) Give the definition of an eigenvalue of a linear map.

(b)  (6 points) Let V be a vector space over F and T ∈ L(V). Prove that T is injective iff 0 is not an eigenvalue of T.