i i i “main” 2007/2/16 page 360 i 360 CHAPTER 5 Linear Transformations For Problems 5–12, describe the transformation of R2 with the given matrix as a product of reﬂections, stretches, and shears. 1 2 5. A = . 0 1 6. A = 10 . 3 1 −1 0 . 0 −1 7. A = 8. A = 9. A = 0 2 . 2 0 1 2 . 3 4 1 0 11. A = . 0 −2 −1 −1 12. A = . −1 0 10. A = 13. Consider the transformation of R2 corresponding to a counterclockwise rotation through angle θ (0 ≤ θ < 2π). For θ = π/2, 3π/2, verify that the matrix of the transformation is given by (5.2.5), and describe it in terms of reﬂections, stretches, and shears. 14. Express the transformation of R2 corresponding to a counterclockwise rotation through an angle θ = π/2 as a product of reﬂections, stretches, and shears. Repeat for the case θ = 3π/2. 1 −3 . −2 8 5.3 The Kernel and Range of a Linear Transformation If T : V → W is any linear transformation, there is an associated homogeneous linear vector equation, namely, T (v) = 0. The solution set to this vector equation is a subset of V called the kernel of T . DEFINITION 5.3.1 Let T : V → W be a linear transformation. The set of all vectors v ∈ V such that T (v) = 0 is called the kernel of T and is denoted Ker(T ). Thus, Ker(T ) = {v ∈ V : T (v) = 0}. Example 5.3.2 Determine Ker(T ) for the linear transformation T : C 2 (I ) → C 0 (I ) deﬁned by T (y) = y + y. Solution: We have Ker(T ) = {y ∈ C 2 (I ) : T (y) = 0} = {y ∈ C 2 (I ) : y + y = 0 for all x ∈ I }. Hence, in this case, Ker(T ) is the solution set to the differential equation y + y = 0. Since this differential equation has general solution y(x) = c1 cos x + c2 sin x, we have Ker(T ) = {y ∈ C 2 (I ) : y(x) = c1 cos x + c2 sin x}. This is the subspace of C 2 (I ) spanned by {cos x, sin x}. The set of all vectors in W that we map onto when T is applied to all vectors in V is called the range of T . We can think of the range of T as being the set of function output values. A formal deﬁnition follows. i i i i i i i “main” 2007/2/16 page 361 i 5.3 361 The Kernel and Range of a Linear Transformation DEFINITION 5.3.3 The range of the linear transformation T : V → W is the subset of W consisting of all transformed vectors from V . We denote the range of T by Rng(T ). Thus, Rng(T ) = {T (v) : v ∈ V }. A schematic representation of Ker(T ) and Rng(T ) is given in Figure 5.3.1. V Ker(T) 0 W Rng(T) Every vector in Ker(T) is mapped to the zero vector in W 0 Figure 5.3.1: Schematic representation of the kernel and range of a linear transformation. Let us now focus on matrix transformations, say T : Rn → Rm . In this particular case, Ker(T ) = {x ∈ Rn : T (x) = 0}. If we let A denote the matrix of T , then T (x) = Ax, so that Ker(T ) = {x ∈ Rn : Ax = 0}. Consequently, If T : Rn → Rm is the linear transformation with matrix A, then Ker(T ) is the solution set to the homogeneous linear system Ax = 0. In Section 4.3, we deﬁned the solution set to Ax = 0 to be nullspace(A). Therefore, we have Ker(T ) = nullspace(A), (5.3.1) from which it follows directly that2 Ker(T ) is a subspace of Rn . Furthermore, for a linear transformation T : Rn → Rm , Rng(T ) = {T (x) : x ∈ Rn }. If A = [a1 , a2 , . . . , an ] denotes the matrix of T , then Rng(T ) = {Ax : x ∈ Rn } = {x1 a1 + x2 a2 + · · · + xn an : x1 , x2 , . . . , xn ∈ R} = colspace(A). Consequently, Rng(T ) is a subspace of Rm . We illustrate these results with an example. Example 5.3.4 Let T : R3 → R2 be the linear transformation with matrix A= 1 −2 5 . −2 4 −10 2 It is also easy to verify this fact directly by using Deﬁnition 5.3.1 and Theorem 4.3.2. i i i i i i i “main” 2007/2/16 page 362 i 362 CHAPTER 5 Linear Transformations Determine Ker(T ) and Rng(T ). Solution: To determine Ker(T ), (5.3.1) implies that we need to ﬁnd the solution set to the system Ax = 0. The reduced row-echelon form of the augmented matrix of this system is 1 −2 5 0 , 0 0 0 0 so that there are two free variables. Setting x2 = r and x3 = s, it follows that x1 = 2r−5s, so that x = (2r − 5s, r, s). Hence, Ker(T ) = {x ∈ R3 : x = (2r − 5s, r, s) : r, s ∈ R} = {x ∈ R3 : x = r(2, 1, 0) + s(−5, 0, 1), r, s ∈ R}. We see that Ker(T ) is the two-dimensional subspace of R3 spanned by the linearly independent vectors (2, 1, 0) and (−5, 0, 1), and therefore it consists of all points lying on the plane through the origin that contains these vectors. We leave it as an exercise to verify that the equation of this plane is x1 − 2x2 + 5x3 = 0. The linear transformation T maps all points lying on this plane to the zero vector in R2 . (See Figure 5.3.2.) x3 y2 2y1 x1 2x2 5x3 0 y2 Rng(T) ) r(T e K y1 x2 T(x) x x1 Figure 5.3.2: The kernel and range of the linear transformation in Example 5.3.6. Turning our attention to Rng(T ), recall that, since T is a matrix transformation, Rng(T ) = colspace(A). From the foregoing reduced row-echelon form of A we see that colspace(A) is generated by the ﬁrst column vector in A. Consequently, Rng(T ) = {y ∈ R2 : y = r(1, −2), r ∈ R}. Hence, the points in Rng(T ) lie along the line through the origin in R2 whose direction is determined by v = (1, −2). The Cartesian equation of this line is y2 = −2y1 . Consequently, T maps all points in R3 onto this line, and therefore Rng(T ) is a onedimensional subspace of R2 . This is illustrated in Figure 5.3.2. To summarize, any matrix transformation T : Rn → Rm with m × n matrix A has natural subspaces Ker(T ) = nullspace(A) Rng(T ) = colspace(A) (subspace of Rn ) (subspace of Rm ) Now let us return to arbitrary linear transformations. The preceding discussion has shown that both the kernel and range of any linear transformation from Rn to Rm are i i i i i i i “main” 2007/2/16 page 363 i 5.3 The Kernel and Range of a Linear Transformation 363 subspaces of Rn and Rm , respectively. Our next result, which is fundamental, establishes that this is true in general. Theorem 5.3.5 If T : V → W is a linear transformation, then 1. Ker(T ) is a subspace of V . 2. Rng(T ) is a subspace of W . Proof In this proof, we once more denote the zero vector in W by 0W . Both Ker(T ) and Rng(T ) are necessarily nonempty, since, as we veriﬁed in Section 5.1, any linear transformation maps the zero vector in V to the zero vector in W . We must now establish that Ker(T ) and Rng(T ) are both closed under addition and closed under scalar multiplication in the appropriate vector space. 1. If v1 and v2 are in Ker(T ), then T (v1 ) = 0W and T (v2 ) = 0W . We must show that v1 + v2 is in Ker(T ); that is, T (v1 + v2 ) = 0W . But we have T (v1 + v2 ) = T (v1 ) + T (v2 ) = 0W + 0W = 0W , so that Ker(T ) is closed under addition. Further, if c is any scalar, T (cv1 ) = cT (v1 ) = c0W = 0W , which shows that cv1 is in Ker(T ), and so Ker(T ) is also closed under scalar multiplication. Thus, Ker(T ) is a subspace of V . 2. If w1 and w2 are in Rng(T ), then w1 = T (v1 ) and w2 = T (v2 ) for some v1 and v2 in V . Thus, w1 + w2 = T (v1 ) + T (v2 ) = T (v1 + v2 ). This says that w1 + w2 arises as an output of the transformation T ; that is, w1 + w2 is in Rng(T ). Thus, Rng(T ) is closed under addition. Further, if c is any scalar, then cw1 = cT (v1 ) = T (cv1 ), so that cw1 is the transform of cv1 , and therefore cw1 is in Rng(T ). Consequently, Rng(T ) is a subspace of W . Remark We can interpret the ﬁrst part of the preceding theorem as telling us that if T is a linear transformation, then the solution set to the corresponding linear homogeneous problem T (v) = 0 is a vector space. Consequently, if we can determine the dimension of this vector space, then we know how many linearly independent solutions are required to build every solution to the problem. This is the formulation for linear problems that we have been looking for. Example 5.3.6 Find Ker(S), Rng(S), and their dimensions for the linear transformation S : M2 (R) → M2 (R) deﬁned by S(A) = A − AT . i i i i i i i “main” 2007/2/16 page 364 i 364 CHAPTER 5 Linear Transformations Solution: In this case, Ker(S) = {A ∈ M2 (R) : S(A) = 0} = {A ∈ M2 (R) : A − AT = 02 }. Thus, Ker(S) is the solution set of the matrix equation A − AT = 02 , so that the matrices in Ker(S) satisfy AT = A. Hence, Ker(S) is the subspace of M2 (R) consisting of all symmetric 2 × 2 matrices. We have shown previously that a basis for this subspace is 1 0 0 1 0 0 , , , 0 0 1 0 0 1 so that dim[Ker(S)] = 3. We now determine the range of S: Rng(S) = {S(A) : A ∈ M2 (R)} = {A − AT : A ∈ M2 (R)} a b a c = − : a, b, c, d ∈ R c d b d 0 b−c = : b, c ∈ R . −(b − c) 0 Thus, Rng(S) = 0 e −e 0 0 1 : e ∈ R = span . −1 0 Consequently, Rng(S) consists of all skew-symmetric 2 × 2 matrices with real elements. Since Rng(S) is generated by the single nonzero matrix 0 1 , −1 0 it follows that a basis for Rng(S) is 0 1 −1 0 , so that dim[Rng(S)] = 1. Example 5.3.7 Let T : P1 → P2 be the linear transformation deﬁned by T (a + bx) = (2a − 3b) + (b − 5a)x + (a + b)x 2 . Find Ker(T ), Rng(T ), and their dimensions. Solution: From Deﬁnition 5.3.1, Ker(T ) = {p ∈ P1 : T (p) = 0} = {a + bx : (2a − 3b) + (b − 5a)x + (a + b)x 2 = 0 for all x} = {a + bx : a + b = 0, b − 5a = 0, 2a − 3b = 0}. i i i i i i i “main” 2007/2/16 page 365 i 5.3 The Kernel and Range of a Linear Transformation 365 But the only values of a and b that satisfy the conditions a + b = 0, b − 5a = 0, 2a − 3b = 0 are a = b = 0. Consequently, Ker(T ) contains only the zero polynomial. Hence, we write Ker(T ) = {0}. It follows from Deﬁnition 4.6.7 that dim[Ker(T )] = 0. Furthermore, Rng(T ) = {T (a + bx) : a, b ∈ R} = {(2a − 3b) + (b − 5a)x + (a + b)x 2 : a, b ∈ R}. To determine a basis for Rng(T ), we write this as Rng(T ) = {a(2 − 5x + x 2 ) + b(−3 + x + x 2 ) : a, b ∈ R} = span{2 − 5x + x 2 , −3 + x + x 2 }. Thus, Rng(T ) is spanned by p1 (x) = 2 − 5x + x 2 and p2 (x) = −3 + x + x 2 . Since p1 and p2 are not proportional to one another, they are linearly independent on any interval. Consequently, a basis for Rng(T ) is {2 − 5x + x 2 , −3 + x + x 2 }, so that dim[Rng(T )] = 2. The General Rank-Nullity Theorem In concluding this section, we consider a fundamental theorem for linear transformations T : V → W that gives a relationship between the dimensions of Ker(T ), Rng(T ), and V . This is a generalization of the Rank-Nullity Theorem considered in Section 4.9. The theorem here, Theorem 5.3.8, reduces to the previous result, Theorem 4.9.1, in the case when T is a linear transformation from Rn to Rm with m × n matrix A. Suppose that dim[V ] = n and that dim[Ker(T )] = k. Then k-dimensions worth of the vectors in V are all mapped onto the zero vector in W . Consequently, we only have n − k dimensions worth of vectors left to map onto the remaining vectors in W . This suggests that dim[Rng(T )] = dim[V ] − dim[Ker(T )]. This is indeed correct, although a rigorous proof is somewhat involved. We state the result as a theorem here. Theorem 5.3.8 (General Rank-Nullity Theorem) If T : V → W is a linear transformation and V is ﬁnite-dimensional, then dim[Ker(T )] + dim[Rng(T )] = dim[V ]. Before presenting the proof of this theorem, we give a few applications and examples. The general Rank-Nullity Theorem is useful for checking that we have the correct dimensions when determining the kernel and range of a linear transformation. Furthermore, it can also be used to determine the dimension of Rng(T ), once we know the dimension of Ker(T ), or vice versa. For example, consider the linear transformation discussed in Example 5.3.6. Theorem 5.3.8 tells us that dim[Ker(S)] + dim[Rng(S)] = dim[M2 (R)], i i i i i i i “main” 2007/2/16 page 366 i 366 CHAPTER 5 Linear Transformations so that once we have determined that dim[Ker(S)] = 3, it immediately follows that 3 + dim[Rng(S)] = 4 so that dim[Rng(S)] = 1. As another illustration, consider the matrix transformation in Example 5.3.4 with 1 −2 5 A= . −2 4 −10 By inspection, we can see that dim[Rng(T )] = dim[colspace(A)] = 1, so the RankNullity Theorem implies that dim[Ker(T )] = 3 − 1 = 2, with no additional calculation. Of course, to obtain a basis for Ker(T ), it becomes necessary to carry out the calculations presented in Example 5.3.4. We close this section with a proof of Theorem 5.3.8. Proof of Theorem 5.3.8: Suppose that dim[V ] = n. We consider three cases: Case 1: If dim[Ker(T )] = n, then by Corollary 4.6.14 we conclude that Ker(T ) = V . This means that T (v) = 0 for every vector v ∈ V . In this case Rng(T ) = {T (v) : v ∈ V } = {0}, hence dim[Rng(T )] = 0. Thus, we have dim[Ker(T )] + dim[Rng(T )] = n + 0 = n = dim[V ], as required. Case 2: Assume dim[Ker(T )] = k, where 0 < k < n. Let {v1 , v2 , . . . , vk } be a basis for Ker(T ). Then, using Theorem 4.6.17, we can extend this basis to a basis for V , which we denote by {v1 , v2 , . . . , vk , vk+1 , . . . , vn }. We prove that {T (vk+1 ), T (vk+2 ), . . . , T (vn )} is a basis for Rng(T ). To do this, we ﬁrst prove that {T (vk+1 ), T (vk+2 ), . . . , T (vn )} spans Rng(T ). Let w be any vector in Rng(T ). Then w = T (v), for some v ∈ V . Using the basis for V , we have v = c1 v1 + c2 v2 + · · · + cn vn for some scalars c1 , c2 , . . . , cn . Hence, w = T (v) = T (c1 v1 + c2 v2 + · · · + cn vn ) = c1 T (v1 ) + c2 T (v2 ) + · · · + cn T (vn ). Since v1 , v2 , . . . , vk are in Ker(T ), this reduces to w = ck+1 T (vk+1 ) + ck+2 T (vk+2 ) + · · · + cn T (vn ). Thus, Rng(T ) = span{T (vk+1 ), T (vk+2 ), . . . , T (vn )}. Next we show that {T (vk+1 ), T (vk+2 ), . . . , T (vn )} is linearly independent. Suppose that dk+1 T (vk+1 ) + dk+2 T (vk+2 ) + · · · + dn T (vn ) = 0, (5.3.2) i i i i i i i “main” 2007/2/16 page 367 i 5.3 The Kernel and Range of a Linear Transformation 367 where dk+1 , dk+2 , . . . , dn are scalars. Then, using the linearity of T , T (dk+1 vk+1 + dk+2 vk+2 + · · · + dn vn ) = 0, which implies that the vector dk+1 vk+1 + dk+2 vk+2 + · · · + dn vn is in Ker(T ). Consequently, there exist scalars d1 , d2 , . . . , dk such that dk+1 vk+1 + dk+2 vk+2 + · · · + dn vn = d1 v1 + d2 v2 + · · · + dk vk , which means that d1 v1 + d2 v2 + · · · + dk vk − (dk+1 vk+1 + dk+2 vk+2 + · · · + dn vn ) = 0. Since the set of vectors {v1 , v2 , . . . , vk , vk+1 , . . . , vn } is linearly independent, we must have d1 = d2 = · · · = dk = dk+1 = · · · = dn = 0. Thus, from Equation (5.3.2), {T (vk+1 ), T (vk+2 ), . . . , T (vn )} is linearly independent. By the work in the last two paragraphs, {T (vk+1 ), T (vk+2 ), . . . , T (vn )} is a basis for Rng(T ). Since there are n − k vectors in this basis, it follows that dim[Rng(T )] = n − k. Consequently, dim[Ker(T )] + dim[Rng(T )] = k + (n − k) = n = dim[V ], as required. Case 3: If dim[Ker(T )] = 0, then Ker(T ) = {0}. Let {v1 , v2 , . . . , vn } be any basis for V . By a similar argument to that used in Case 2 above, it can be shown that (see Problem 18) {T (v1 ), T (v2 ), . . . , T (vn )} is a basis for Rng(T ), and so again we have dim[Ker(T )] + dim[Rng(T )] = n. Exercises for 5.3 Key Terms Kernel and range of a linear transformation, Rank-Nullity Theorem. Skills • Be able to ﬁnd the kernel of a linear transformation T : V → W and give a basis and the dimension of Ker(T ). • Be able to ﬁnd the range of a linear transformation T : V → W and give a basis and the dimension of Rng(T ). • Be able to show that the kernel (resp. range) of a linear transformation T : V → W is a subspace of V (resp. W ). • Be able to verify the Rank-Nullity Theorem for a given linear transformation T : V → W . • Be able to utilize the Rank-Nullity Theorem to help ﬁnd the dimensions of the kernel and range of a linear transformation T : V → W . True-False Review For Questions 1–6, decide if the given statement is true or false, and give a brief justiﬁcation for your answer. If true, you can quote a relevant deﬁnition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is false. 1. If T : V → W is a linear transformation and W is ﬁnite-dimensional, then dim[Ker(T )] + dim[Rng(T )] = dim[W ]. 2. If T : P4 → R7 is a linear transformation, then Ker(T ) must be at least two-dimensional. i i i i i i i “main” 2007/2/16 page 368 i 368 CHAPTER 5 Linear Transformations 3. If T : Rn → Rm is a linear transformation with matrix A, then Rng(T ) is the solution set to the homogeneous linear system Ax = 0. 4. The range of a linear transformation T : V → W is a subspace of V . 5. If T : M23 → P7 is a linear transformation with 1 1 1 1 2 3 T = 0, T = 0, 0 0 0 4 5 6 then Rng(T ) is at most four-dimensional. 6. If T : Rn → Rm is a linear transformation with matrix A, then Rng(T ) is the column space of A. Problems 1. Consider T : R3 → R2 deﬁned by T (x) = Ax, where 1 −1 2 A . 1 −2 −3 Find T (x) and thereby determine whether x is in Ker(T ). (a) x = (7, 5, −1). (b) x = (−21, −15, 2). (c) x = (35, 25, −5). For Problems 2–6, ﬁnd Ker(T ) and Rng(T ) and give a geometrical description of each. Also, ﬁnd dim[Ker(T )] and dim[Rng(T )] and verify Theorem 5.3.8. 2. T : R2 → R2 deﬁned by T (x) = Ax, where 3 6 A= . 1 2 3. T : R3 → R3 deﬁned by T (x) = Ax, where 1 −1 0 A = 0 1 2. 2 −1 1 6. T : R3 → R2 deﬁned by T (x) = Ax, where 1 3 2 A= . 2 6 5 For Problems 7–10, compute Ker(T ) and Rng(T ). 7. The linear transformation T deﬁned in Problem 24 in Section 5.1. 8. The linear transformation T deﬁned in Problem 25 in Section 5.1. 9. The linear transformation T deﬁned in Problem 26 in Section 5.1. 10. The linear transformation T deﬁned in Problem 27 in Section 5.1. 11. Consider the linear transformation T : R3 → R deﬁned by T (v) = u, v, where u is a ﬁxed nonzero vector in R3 . (a) Find Ker(T ) and dim[Ker(T )], and interpret this geometrically. (b) Find Rng(T ) and dim[Rng(T )]. 12. Consider the linear transformation S : Mn (R) → Mn (R) deﬁned by S(A) = A + AT , where A is a ﬁxed n × n matrix. (a) Find Ker(S) dim[Ker(S)]? and describe it. What is (b) In the particular case when A is a 2 × 2 matrix, determine a basis for Ker(S), and hence, ﬁnd its dimension. 13. Consider the linear transformation T : Mn (R) → Mn (R) deﬁned by T (A) = AB − BA, where B is a ﬁxed n × n matrix. Find Ker(T ) and describe it. 4. T : R3 → R3 deﬁned by T (x) = Ax, where 1 −2 1 A = 2 −3 −1 . 5 −8 −1 14. Consider the linear transformation T : P2 → P2 deﬁned by 5. T : R3 → R2 deﬁned by T (x) = Ax, where 1 −1 2 A= . −3 3 −6 (a) Show that Ker(T ) consists of all polynomials of the form b(x − 2), and hence, ﬁnd its dimension. T (ax 2 +bx +c) = ax 2 +(a+2b+c)x +(3a−2b−c), where a, b, and c are arbitrary constants. (b) Find Rng(T ) and its dimension. i i i i i i i “main” 2007/2/16 page 369 i 5.4 Additional Properties of Linear Transformations 15. Consider the linear transformation T : P2 → P1 deﬁned by 369 T (v3 ) = w1 + 2w2 . Find Ker(T ), Rng(T ), and their dimensions. T (ax 2 + bx + c) = (a + b) + (b − c)x, where a, b, and c are arbitrary real numbers. Determine Ker(T ), Rng(T ), and their dimensions. 18. (a) Let T : V → W be a linear transformation, and suppose that dim[V ] = n. If Ker(T ) = {0} and {v1 , v2 , . . . , vn } is any basis for V , prove that 16. Consider the linear transformation T : P1 → P2 deﬁned by T (ax + b) = (b − a) + (2b − 3a)x + bx 2 . {T (v1 ), T (v2 ), . . . , T (vn )} Determine Ker(T ), Rng(T ), and their dimensions. is a basis for Rng(T ). (This ﬁlls in the missing details in the proof of Theorem 5.3.8.) 17. Let {v1 , v2 , v3 } and {w1 , w2 } be bases for real vector spaces V and W , respectively, and let T : V → W be the linear transformation satisfying T (v1 ) = 2w1 − w2 , 5.4 (b) Show that the conclusion from part (a) fails to hold if Ker(T ) = {0}. T (v2 ) = w1 − w2 , Additional Properties of Linear Transformations One aim of this section is to establish that all real vector spaces of (ﬁnite) dimension n are essentially the same as Rn . In order to do so, we need to consider the composition of linear transformations. DEFINITION 5.4.1 Let T1 : U → V and T2 : V → W be two linear transformations.3 We deﬁne the composition, or product, T2 T1 : U → W by (T2 T1 )(u) = T2 (T1 (u)) for all u ∈ U. The composition is illustrated in Figure 5.4.1. U W T2T1 u T1 V (T2T1)(u) T2(T1(u)) T2 T1(u) Figure 5.4.1: The composition of two linear transformations. Our ﬁrst result establishes that T2 T1 is a linear transformation. Theorem 5.4.2 Let T1 : U → V and T2 : V → W be linear transformations. Then T2 T1 : U → W is a linear transformation. 3 We assume that U, V , and W are either all real vector spaces or all complex vector spaces. i i i i

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