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. Getting Started: This is an "if and only if" statement, so you need to prove the statement in both directions. To prove that Tis a linear transformation, you need to show that the function satisfies the definition of a linear transformation. In the other direction, let T be a linear transformation. Use the definition and properties of a linear transformation to prove that T(au + bv) - aT(u) + bT(v). (1) Let T(au + bv) = aT(u) + bT(v). Show that T preserves the properties of vector addition and scalar multiplication by choosing appropriate 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 that T(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 for our 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 Vand let a- b= 1. O Let u - v and let a and b be arbitrary scalars. O 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. O Let u =v= 0 and let a and b be arbitrary scalars. O Let u and v be arbitrary vectors in Vand 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 and v be arbitrary vectors in V and let a = b. O Let u be an arbitrary vector in V, v = 0, and let a and b be arbitrary scalars. O Let u-v and let a = b = 0.5. O Let u- v -0 and let a -b- 1. 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 Tis a linear transformation. Use the properties and definition of a linear transformation to show that T(au + bv) = aT(u) + bT(v). v. So by the properties of linear transformations we have the 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 au and bv are -Select-- following. T(au + bv) = -Select--- V- aT(u) + bT(v)

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. Getting Started: This is an "if and only if" statement, so you need to prove the statement in both directions. To prove that Tis a linear transformation, you need to show that the function satisfies the definition of a linear transformation. In the other direction, let T be a linear transformation. Use the definition and properties of a linear transformation to prove that T(au + bv) - aT(u) + bT(v). (1) Let T(au + bv) = aT(u) + bT(v). Show that T preserves the properties of vector addition and scalar multiplication by choosing appropriate 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 that T(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 for our 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 Vand let a- b= 1. O Let u - v and let a and b be arbitrary scalars. O 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. O Let u =v= 0 and let a and b be arbitrary scalars. O Let u and v be arbitrary vectors in Vand 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 and v be arbitrary vectors in V and let a = b. O Let u be an arbitrary vector in V, v = 0, and let a and b be arbitrary scalars. O Let u-v and let a = b = 0.5. O Let u- v -0 and let a -b- 1. 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 Tis a linear transformation. Use the properties and definition of a linear transformation to show that T(au + bv) = aT(u) + bT(v). v. So by the properties of linear transformations we have the 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 au and bv are -Select-- following. T(au + bv) = -Select--- V- aT(u) + bT(v)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.6: Introduction To Linear Transformations

Problem 26EQ

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