Exercises – Section 2.1: Linear Transformations, Null
Spaces, and Ranges
1. Label the following statements as true or false. Let V and W be finite-dimensional
vector spaces over F , and T : V → W .
(a) If T is linear, then T preserves sums and scalar products.
(b) If T (x + y) = T (x) + T (y), then T is linear.
(c) T is one-to-one if and only if the only vector x such that T (x) = 0 is x = 0.
(d) If T is linear, then T (0V ) = 0W .
(e) If T is linear, then nullity(T ) + rank(T ) = dim(W ).
(f) If T is linear, then T maps linearly independent subsets of V onto linearly independent subsets of W .
(g) If T, U : V → W are linear and agree on a basis for V , then T = U .
(h) Given x1 , x2 ∈ V and y1 , y2 ∈ W , there exists a linear transformation T : V → W
such that T (x1 ) = y1 and T (x2 ) = y2 .
2. –
6. For Exercises 2 through 6, prove that T is a linear transformation, and find bases for
both N (T ) and R(T ). Then compute the nullity and rank of T , verify the dimension
theorem, and determine whether T is one-to-one or onto.
(a) T : R3 → R2 defined by T (a1 , a2 , a3 ) = (a1 − a2 , 2a3 ).
(b) T : R2 → R3 defined by T (a1 , a2 ) = (a1 + a2 , 0, 2a1 − a2 ).
(c) T : M2×3 (F ) → M2×2 (F ) defined by
a11 a12 a13
2a11 − a12 a13 + 2a12
T
=
a21 a22 a23
0
0
(d) T : P2 (R) → P3 (R) defined by T (f (x)) = xf (x) + f ′ (x).
(e) T : Mn×n (F ) → F defined by T (A) = tr(A).
3. Prove properties 1–4 from page 65.
4. Prove that the transformations in Examples 2 and 3 are linear.
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5. Identify why each T : R2 → R2 is not linear.
(a) T (a1 , a2 ) = (1, a2 )
(b) T (a1 , a2 ) = (a1 , a21 )
(c) T (a1 , a2 ) = (sin a1 , 0)
(d) T (a1 , a2 ) = (|a1 |, a2 )
(e) T (a1 , a2 ) = (a1 + 1, a2 )
6. Let T : R2 → R2 be linear with T (1, 0) = (1, 4) and T (1, 1) = (2, 5). Find T (2, 3) and
determine whether T is one-to-one.
7. Prove that there exists a linear transformation T : R2 → R3 such that T (1, 1) = (1, 0, 2)
and T (2, 3) = (1, −1, 4). Find T (8, 11).
8. Determine whether a linear transformation T : R3 → R2 exists with T (1, 0, 3) = (1, 1)
and T (−2, 0, −6) = (2, 1).
9. Let T : V → W be linear. If {w1 , . . . , wk } ⊂ R(T ) is linearly independent and
T (vi ) = wi , prove {v1 , . . . , vk } is linearly independent.
10. Let T : V → W be linear.
(a) Prove T is one-to-one iff T carries linearly independent subsets of V onto linearly
independent subsets of W .
(b) Suppose T is one-to-one and S ⊂ V . Then S is linearly independent iff T (S) is.
(c) Let β = {v1 , . . . , vn } be a basis. If T is an isomorphism, prove T (β) is a basis.
Rx
11. Define T (f (x)) = 0 f (t)dt on P (R). Prove T is linear and one-to-one but not onto.
12. Let T (f (x)) = f ′ (x) on P (R). Prove T is onto but not one-to-one.
13. Let T : V → W be linear with dim V < dim W . Prove T is not onto. If dim V >
dim W , prove T is not one-to-one.
14. Give an example of T : R2 → R2 such that N (T ) = R(T ).
15. Give examples of distinct linear maps T, U such that N (T ) = N (U ) and R(T ) = R(U ).
16. Prove that the image of a subspace under a linear map is a subspace, and that preimages
of subspaces are subspaces.
17. Let T, U : V → V be left and right shift operators on sequence space. Prove:
(a) T , U are linear.
(b) T is onto but not one-to-one.
(c) U is one-to-one but not onto.
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18. Let T : R3 → R be linear. Show there exist scalars a, b, c such that T (x, y, z) =
ax + by + cz. Generalize to T : F n → F m .
19. Geometrically describe the null space of T : R3 → R linear.
20. Definition: If V = W1 ⊕ W2 , a projection T : V → V onto W1 along W2 satisfies
T (x) = x1 for x = x1 + x2 , x1 ∈ W1 , x2 ∈ W2 .
21. Let T : R2 → R2 be projection on the y-axis:
(a) Along x-axis.
(b) Along line L = {(s, s) : s ∈ R}.
22. Similar projection problems in R3 :
(a) T (a, b, c) = (a, b, 0) is projection onto xy-plane along z-axis.
(b) Find projection onto z-axis along xy-plane.
(c) T (a, b, c) = (a − c, b, 0) is projection onto xy-plane along L = {(a, 0, a)}.
23. For a projection T , prove:
(a) T is linear and W1 = {x ∈ V : T (x) = x}.
(b) R(T ) = W1 , N (T ) = W2 .
(c) Describe T when W1 = V and W1 = {0}.
24. Show for any subspace W in finite-dimensional V there exists T projecting onto W
along some W ′ .
25. Define T -invariance. Let T : V → V be linear. Prove:
(a) {0}, V, R(T ), N (T ) are T -invariant.
(b) If W is T -invariant, then TW is linear.
(c) If T is a projection onto W along W ′ , then TW = IW .
(d) If V = R(T ) ⊕ W and W is T -invariant, then W ⊆ N (T ) and if dim V < ∞, then
W = N (T ).
(e) If W is T -invariant, then N (TW ) = N (T ) ∩ W and R(TW ) = T (W ).
26. Prove Theorem 2.2 for infinite basis β: R(T ) = span{T (v) : v ∈ β}.
27. Prove generalization of Theorem 2.6: For any function f : β → W from a basis β of
V , there exists a unique linear map T such that T (x) = f (x) for all x ∈ β.
28. Let V be finite-dimensional and T : V → V linear.
(a) If V = R(T ) + N (T ), prove V = R(T ) ⊕ N (T ).
(b) If R(T ) ∩ N (T ) = {0}, then V = R(T ) ⊕ N (T ).
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29. Let V, T as in shift operator example.
(a) Prove V = R(T ) + N (T ) but not a direct sum.
(b) Find T1 such that R(T1 ) ∩ N (T1 ) = {0} but not direct sum.
30. Define an additive function T : V → W by T (x + y) = T (x) + T (y). Prove that if V, W
are vector spaces over Q, then additive implies linear.
31. Let T : C → C, T (z) = z̄. Prove T is additive but not linear.
32. Construct an additive T : R → R not linear (using R over Q and basis arguments).
33. Let η : V → V /W be defined by η(v) = v + W . Prove:
(a) η is linear and onto, N (η) = W .
(b) If dim V < ∞, derive formula dim(V ) = dim(W ) + dim(V /W ).
(c) Compare to the proof from Exercise 35, Section 1.6.
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