MATH 235 W24- A5 Question 1 Part I (a) and (b) only to be marked. Let V, W, and X be vector spaces. Let T₁: V → W and T₂: W→ X be linear transformations. Consider the composite linear transformation T₂ o T₁: V → X, defined by (T₂T₁)(v) = T₂(T₁(v)), Vv € V. Part I (a) Show that if both T₁ and T₂ are onto, then T₂ o T₁ is also onto. (b) Show that if T₂o T₁ is onto, then T₂ is onto. (c) Provide a counterexample to show that if T₂ oT₁ is onto, then 7₁ need not be onto. In your counterexample, clearly state the range of T₁. Part II (a) Show that if both T₁ and T₂ are one-to-one, then T₂ o This also one-to-one. (b) Show that if T₂ o T₁ is one-to-one, then T₁ is one-to-one. (c) Provide a counterexample to show that if T₂ T₁ is one-to-one, then 7₂ need not be one-to-one. In your counterexample, clearly state the nullspace of T₂-

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