Let U, V, W be vector spaces over the same field, and let f, g be...

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Let U, V, W be vector spaces over the same field, and let f, g be linear maps as follows: Uv4 w, with the property that gof = 0. 1. Prove that Im(f) C Null(g), and give an explicit example of vector spaces and linear maps as above, with the property that the Im(f) is a proper subspace of Null(g) and Null(g) is a proper subspace of V. 2. Suppose that Q : V → V is a linear operator on a finite dimensional vector space, and suppose Q? = 0. Suppose further that Null(Q) = Im(Q). Prove that dim V must be even. Provide an example of such a linear map. Let U, V, W be vector spaces over the same field, and let f, g be linear maps as follows: Uv4 w, with the property that gof = 0. 1. Prove that Im(f) C Null(g), and give an explicit example of vector spaces and linear maps as above, with the property that the Im(f) is a proper subspace of Null(g) and Null(g) is a proper subspace of V. 2. Suppose that Q : V → V is a linear operator on a finite dimensional vector space, and suppose Q? = 0. Suppose further that Null(Q) = Im(Q). Prove that dim V must be even. Provide an example of such a linear map.