MAT A22 Winter 2024 Homework # 7
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Homework # 7
MAT A22
Winter 2024
Department of Computer and Mathematical Sciences
Homework Guidelines
This homework was released on Fri. Mar. 1st 14:00 (EST). It is due on Fri. Mar. 8th 17:00 (EST).
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Readings
❼ ➜2.3 Kernel and Image
Problems
Recall, from the Week 4 Tutorial Activity: if U, W ⊆ V are subspaces such that every vector in v ∈ U + W can be written uniquely w = u + w with u ∈ U and w ∈ W then we say that the sum U + W is direct and write U + W = U ⊕ W.
Let T : V → V. Suppose we have an invariant subspace W, then we can restrict ourselves to simply looking at the subspace W and define the restriction operator TW of T onto W. The restriction operator TW : W → W is a linear transformation such that TW (v) = T(v) for all v ∈ W.
Let T : V → V. Then T2 is defined as T2 (v) = T(T(v)). Later we will prove that the composition linear transformations forms a linear transformation.
Q1. Suppose that V is finite dimensional and let T : V → V be a linear map. In the previous assignment, we proved that the first three of these five statements are equivalent. Prove that all 5 statements are equivalent.
(a) image(T2 ) = image(T)
(b) ker(T) = ker(T2 ).
(c) image(T) ∩ ker(T) = {0}.
(d) V = image(T) ⊕ ker(T).
(e) ker (Timage(T), = {0}.
Q2. Suppose that W1 and W2 are subspaces of V with dim(W1 ) = n, dim(W2 ) = k, and n ≤ k. Prove the following:
(a) dim(W1 ∩ W2 ) ≤ n
(b) dim(W1 + W2 ) ≤ n + k
Prove examples of W1 and W2 ⊆ R3 such that n < k and:
(a) dim(W1 ∩ W2 ) = n
(b) dim(W1 + W2 ) = n + k
A function is bijective if it is both injective and surjective.
Q3. Given a set of vectors S = {v1 , . . . , vn } consider the coordinate linear map T : Rn → V given by
T(x1 ,..., xn ) = x1 v1 + x2 v2 + ··· + xnvn.
(a) What is the image of T?
(b) What is the rank of T?
(c) Prove that T is bijective if and only if V is a basis for V.
Q4. We will later show that if V is a finite dimensional vector space, then T : V → V is injective if and only
if it is subjective if and only if it is bijection. This is not true for infinite dimensional vector spaces.
Consider the following two operation.
RS(x1 , x2 , x2 , ...) = (0, x1 , x2 , ...) and LS(x1 , x2 , x3 , ...) = (x2 , x3 , ...)
(a) The function RS is called the right-shift function. Show that it is injective but not surjective. (b) The function LS is called the left-shift function. Show that it is surjective but not injective.
Q5. Suppose that V and W are finite dimensional vector space. Prove the following:
(a) There exists an injective linear transformation T : V → W if and only if dim(W) ≥ dim(V). (b) There exists a surjective linear transformation T : V → W if and only if dim(W) ≤ dim(V). (Hint: We proved one of the directions in class together.)
Q6. Suppose that dim(V) = k. Prove the following:
(a) Any k linearly independent vectors form a basis of V.
(b) Any k vectors that span V form a basis of V.
This question shows that, in a finite dimensional vector space, a set of vectors is a basis if it is the “right size” (its size is dim(V)) and either: spans V or is linearly independent.
The following question is a challenge question that will not be graded.
Q7. Suppose that T : V → V is a linear transformation and V is finite dimensional.
(a) Prove that Rank(T) > Rank(T2 ).
(b) Prove that ifRank(Tn ) = Rank(Tn+1), then Rank(Tn ) = Rank(Tn+k) for all finite k > 0.
(c) Suppose that T is a nilpotent operator. What can we say about nullity(Tn+1) ifRank(Tn ) > 0?
(d) Suppose that T is not nilpotent. Show that there exists some operator S = Tk such that V =
image(S) ⊕ ker(S). (Hint: It is easier to prove an equivalent statement from question 1.)
2024-03-11