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Let V and W be finite-dimensional vector spaces over F, and let φ : V →...

Let V and W be finite-dimensional vector spaces over F, and let φ : V → W be a linear transformation. Let dim(ker(φ)) = k, dim(V ) = n, and 0 < k < n. A basis of ker(φ), {v1, . . . , vk}, can be extended to a basis of V , {v1, . . . , vk, vk+1, . . . , vn}, for some vectors vk+1, . . . , vn ∈ V . Prove that {φ(vk+1), . . . , φ(vn)} is a linearly independent set without using the Rank Nullity Theorem.

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