5Prove that T: V→ W is a linear transformation if and only ifT(au + bv) = aT(u) + bT(v)for all vectors u and v and all scalars a and b.Getting Started: This is an "if and only if" statement, so you need to prove the statement in both directions. To prove that T is a lineartransformation, you need to show that the function satisfies the definition of a linear transformation. In the other direction, let T be a lineartransformation. Use the definition and properties of a linear transformation to prove thatT(au + bv) = aT(u) + bT(v).(i) Let T(au + bv) = aT(u) + bT(v). Show that T preserves the properties of vector addition and scalar multiplication by choosingappropriate values of a and b.Let a and b be scalars and u and v be vectors in V. Suppose that for all possible values of a and b and all vectors u and v thatT(au + bv) = aT(u) + bT(v).Since the equality holds for all scalars a and b and all vectors u and v we are free to choose values for each of these quantities forour proof. Which of the following choices can be used to prove the first property of a linear transformation, T(u + v) = T(u) + T(v)?O Let u and v be arbitrary vectors in V and let a = -b.O Let u v and let a and b be arbitrary scalars.O Let u = v= 0 and let a and b be arbitrary scalars.O Let u and v be arbitrary vectors in V and let a = b = 1.○ Let u = v= I and let a and b be arbitrary scalars.O Let u and v be arbitrary vectors in V and let a = b = 0.Similarly, which of the following choices for u, v, a, and b can be used to prove the second property of a linear transformation,T(cu) = cT(u)?O Let u = v and let a = b = 0.5.O Let u be an arbitrary vector in V, v = 0, and let a and b be arbitrary scalars.O Let u = v= 0 and let a = b = 1.O Let u and v be arbitrary vectors in V and let a = b.O Let u be an arbitrary vector in V, v = I, and let a and b be arbitrary scalars.(ii) To prove the statement in the other direction, assume that T is a linear transformation. Use the properties and definition of a lineartransformation to show that T(au + bv) = aT(u) + bT(v).Let T: V → W be a linear transformation, u and v be vectors in V, and a and b be scalars. Since V is a vector space, we know that auand bv are ---Select---So by the properties of linear transformations we have the following.T(au + bv) = ---Select---= aT(u) + bT(v)

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5Prove that T: V→ W is a linear transformation if and only ifT(au + bv) = aT(u) + bT(v)for all vectors u and v and all scalars a and b.Getting Started: This is an &#34;if and only if&#34; statement, so you need to prove the statement in both directions. To prove that T is a lineartransformation, you need to show that the function satisfies the definition of a linear transformation. In the other direction, let T be a lineartransformation. Use the definition and properties of a linear transformation to prove thatT(au + bv) = aT(u) + bT(v).(i) Let T(au + bv) = aT(u) + bT(v). Show that T preserves the properties of vector addition and scalar multiplication by choosingappropriate values of a and b.Let a and b be scalars and u and v be vectors in V. Suppose that for all possible values of a and b and all vectors u and v thatT(au + bv) = aT(u) + bT(v).Since the equality holds for all scalars a and b and all vectors u and v we are free to choose values for each of these quantities forour proof. Which of the following choices can be used to prove the first property of a linear transformation, T(u + v) = T(u) + T(v)?O Let u and v be arbitrary vectors in V and let a = -b.O Let u v and let a and b be arbitrary scalars.O Let u = v= 0 and let a and b be arbitrary scalars.O Let u and v be arbitrary vectors in V and let a = b = 1.○ Let u = v= I and let a and b be arbitrary scalars.O Let u and v be arbitrary vectors in V and let a = b = 0.Similarly, which of the following choices for u, v, a, and b can be used to prove the second property of a linear transformation,T(cu) = cT(u)?O Let u = v and let a = b = 0.5.O Let u be an arbitrary vector in V, v = 0, and let a and b be arbitrary scalars.O Let u = v= 0 and let a = b = 1.O Let u and v be arbitrary vectors in V and let a = b.O Let u be an arbitrary vector in V, v = I, and let a and b be arbitrary scalars.(ii) To prove the statement in the other direction, assume that T is a linear transformation. Use the properties and definition of a lineartransformation to show that T(au + bv) = aT(u) + bT(v).Let T: V → W be a linear transformation, u and v be vectors in V, and a and b be scalars. Since V is a vector space, we know that auand bv are ---Select---So by the properties of linear transformations we have the following.T(au + bv) = ---Select---= aT(u) + bT(v)

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Prove that T: V→ W is a linear transformation if and only if T(au + bv) = aT(u) + bT(v) for all vectors u and v and all scalars a and b.