Chapter 7 Coursework Sheets

7.1 Coursework Sheet 1

Submit a single pdf with scans of your work to Blackboard by Tuesday, 8 February 2022, 17:00.

Exercise 1

Write down a real \((2\times 2)\)-matrix \(A\) and a vector \(\mathbf{b} \in \mathbb{R}^2\), both without zero entries, such that the equation \(A\mathbf{x} = \mathbf{b}\) has

  1. no solution \(\mathbf{x}\) in \(\mathbb{R}^2\).

  2. exactly one solution \(\mathbf{x}\) in \(\mathbb{R}^2\).

  3. infinitely many solutions \(\mathbf{x}\) in \(\mathbb{R}^2\).

(Give reasons for your answers.)

Exercise 2

  1. Let \(w,z\) be complex numbers. Solve the linear equation \(wx=z\); in other words, find all \(x\in\mathbb{C}\) such that \(wx=z\). (Hint: You need to distinguish three cases.)

  2. Solve the following system of linear equations: \[\begin{align*} (5i)x_1 + 3x_2 &= 12+i\\ x_1 - (2i)x_2 &= 3-i. \end{align*}\]

Exercise 3

Let \(A\) be a real \(m\times n\) matrix. Prove that its column space \(\mathrm{Col}(A)\) is a subspace of \(\mathbb{R}^m\). (Recall that, if we denote the columns of \(A\) by \(\mathbf{v}_1,\dots,\mathbf{v}_n\in\mathbb{R}^m\), then the column space of \(A\) is the set of all linear combinations of the vectors \(\mathbf{v}_1,\dots,\mathbf{v}_n\). Symbolically, \(\mathrm{Col}(A) = \{ a_1\mathbf{v}_1+\dots+a_n\mathbf{v}_n \mid a_1,\dots,a_n\in \mathbb{R}\}\subseteq\mathbb{R}^m\).)

Extra question (not assessed)

Multiplication of complex numbers defines a binary operation on \(\mathbb{C}^\times := \mathbb{C}\setminus\{0\}\). Show that \(\mathbb{C}^\times\) together with this operation is an abelian group. (Here we consider the multiplication of complex numbers defined like so: if \(a+bi\) and \(c+di\) (\(a,b,c,d\in\mathbb{R}\)) are complex numbers, their product is declared to be the complex number \((ac-bd) + (ad+bc)i\). In your arguments, you may without further discussion use the usual laws of algebra for \(\mathbb{R}\). such as associativity for addition and multiplication of real numbers.)

Challenge question (not assessed)

Let \(G\) be a group such that for all \(a\in G\) we have \(a*a=e\). Show that \(G\) is abelian.

7.2 Coursework Sheet 2

Submit a single pdf with scans of your work to Blackboard by Tuesday, 15 February 2022, 17:00.

Exercise 1

Let \(G\) and \(H\) be groups with binary operations \(\boxplus\) and \(\odot\), respectively. We define a binary operation \(\ast\) on the cartesian product \(G \times H\) by \[(a,b) \ast (a',b') := (a \boxplus a', b \odot b') \quad (\textrm{for } a,a' \in G \textrm{ and } b,b' \in H).\] Show that \(G\times H\) together with this operation is a group.

Exercise 2

For \(a, b \in \mathbb{R}\) we define \(a \ast b := ab-a-b+2 \in \mathbb{R}\). Furthermore let \(G:= \mathbb{R} \backslash \{1\}\).

  1. Show that \(a \ast b \in G\) for all \(a, b \in G\).

  2. Show that \(G\) together with the binary operation \(G \times G \rightarrow G\), \((a, b) \mapsto a \ast b\), is a group.

Exercise 3

Let \(G=\{s,t,u,v\}\) be a group with \(s\ast u = u\) and \(u\ast u = v\). Determine the group table of \(G\). (There is only one way of completing the group table for \(G\). Give a reason for each step.)

Exercise 4

Write down the group tables for the groups \(C_4\) and \(C_2 \times C_2\) (cf. Exercise 1). For every element \(a\) in \(C_4\) and \(C_2 \times C_2\) determine the smallest positive integer \(m\) such that \(ma\) equals the identity element.

Extra question (not assessed — no need to submit)

Let \(G\) be a group whose binary operation is written additively, i.e. \(G \times G \to G\), \((a,b) \mapsto a+b\). Show that \(m(n a) = (m n) a\) for all \(a \in G\) and \(m,n \in \mathbb{Z}\). (Hint: You need to distinguish up to 9 cases.) Write down the other two exponential laws in additive notation as well. (Formulate these laws as complete mathematical statements including all quantifiers. No proofs are required.)

7.3 Coursework Sheet 3

Submit a single pdf with scans of your work to Blackboard by Tuesday, 22 February 2022, 17:00.

Exercise 1

Write down the group table for the permutation group \(S_3\) and show that \(S_3\) is not abelian. (You may find it more convenient to write all elements of \(S_3\) in cycle notation.)

Exercise 2

\[\textrm{Let}\quad \sigma:= \left(\begin{array}{ccccccccc}1&2&3&4&5&6&7&8&9\\3&4&6&7&8&9&1&2&5\end{array}\right) \in S_9, \qquad \tau := \left(\begin{array}{cccccc}1&2&3&4&5&6\\3&4&5&2&1&6\end{array}\right) \in S_6\] \[\textrm{and} \quad \eta := \left(\begin{array}{ccccc}1& 2& \ldots & n-1 & n \\n& n-1& \ldots & 2 & 1\end{array}\right) \in S_n \quad (\textrm{for any even } n \in \mathbb{N}).\]

  1. Determine the sign of \(\sigma\), \(\tau\) and \(\eta\).

  2. Write \(\sigma^2\), \(\sigma^{-1}\), \(\tau^2\), \(\tau^{-1}\), \(\eta^2\) and \(\eta^{-1}\) as a composition of cycles.

  3. Determine the sign of \(\sigma^2\), \(\tau^2\) and \(\eta^2\) in two ways, firstly using (b) and secondly using (a) and Theorem 1.10 (b).

Exercise 3

Let \(n \ge 1\). Let \(\langle a_1, \ldots, a_s \rangle \in S_n\) be a cycle and let \(\sigma \in S_n\) be arbitrary. Show that \[\sigma \circ \langle a_1, \ldots, a_s\rangle \circ \sigma^{-1} = \langle \sigma(a_1), \ldots, \sigma(a_s)\rangle \textrm{ in } S_n.\] (Note this is an equality between maps. Hence, in order to show this equality you need to show that both sides are equal after applying them to an arbitrary element \(b\) of \(\{1, 2, \ldots, n\}\). To do so you will need to distinguish whether \(b\) belongs to \(\{\sigma(a_1), \ldots, \sigma(a_s)\}\) or not.)

Exercise 4

Denote by \(\sqrt{-3}\) a square root of \(-3\) in \(\mathbb{C}\). Let \(\mathbb{Q}(\sqrt{-3})\) denote the set of complex numbers \(z\) of the form \(z= a + b \sqrt{-3}\) where \(a, b \in \mathbb{Q}\). Show that \(\mathbb{Q}(\sqrt{-3})\) together with the usual addition and multiplication of real numbers is a field. (Hint: You need to show that for any \(w,z \in \mathbb{Q}(\sqrt{-3})\) also \(w+z, wz, -z\) and \(z^{-1}\) (if \(z \not= 0\)) are in \(\mathbb{Q}(\sqrt{-3})\) and that \(0\) and \(1\) are in \(\mathbb{Q}(\sqrt{-3})\). Distributivity, commutativity and associativity for addition and multiplication hold in \(\mathbb{Q}(\sqrt{-3})\) because they hold in \(\mathbb{R}\).)

Exercise 5

Let \(F\) be a field. For any \(a, b \in F\), \(b \not= 0\), we write \(\frac{a}{b}\) for \(ab^{-1}\). Prove the following statements for any \(a, a' \in F\) and \(b,b' \in F\backslash \{0\}\):

  1. \(\displaystyle{\frac{a}{b} + \frac{a'}{b'} = \frac{ab'+a'b}{bb'}}\); (ii) \(\displaystyle{\frac{a}{b}\frac{a'}{b'} = \frac{aa'}{bb'}}\).

Extra items for Exercise 5 (not assessed, do not submit):

  1. \(\displaystyle{\frac{a}{b} = \frac{a'}{b'}}\) if and only if \(ab' = a'b\);

  2. \(\displaystyle{\frac{\frac{a}{b}}{\frac{a'}{b'}} = \frac{ab'}{a'b}}\) (if in addition \(a' \not= 0\)).

Extra problems to think about (do not submit)

The solutions for this will not be provided (but possible to find in a book or google). Not necessary for the rest of the module at all. Feel free to ignore.

Task 1. The aim is to prove Thm 1.10 from the notes, about the sign function on the symmetric groups \(S_n\). Here’s one possible path to a proof.

  • Every cycle of length \(k\) can be written as a product of \(k-1\) transpositions.
  • Thus, every permutation can be written as a product of transpositions.
  • Let \(\sigma\) be a permutation, and write it as a product of transpositions. Define the number \(\mathrm{nsgn}(\sigma)\) (for ``new sign’’) to be equal to \(1\) if the number of transpositions is even, and \(-1\) if the number of transpositions is odd. Again, apriori \(\mathrm{nsgn}\) depends on how do we write \(\sigma\) as a product of transpositions. However, by the first point above, \(\mathrm{nsgn}(\sigma)=\mathrm{sgn}(\sigma)\), since every cycle decomposition of \(\sigma\) gives also a way to write \(\sigma\) as a product of transpositions. So the goal now is to prove that \(\mathrm{nsgn}\) is well defined, and that it’s multiplicative.
  • A way to prove the above is to find a way to characterise \(\mathrm{nsgn}\) to be something intrinsic to a permutation. Here’s such a thing: Given a permutation \(\sigma\in S_n\), we say that \(\sigma\) reverses the pair \((i,j)\), if \(i,j\in\{1,\dots,n\}\), \(i<j\) and \(\sigma(i)>\sigma(j)\). Let \(\mathrm{isgn}(\sigma)\) be \(1\) if \(\sigma\) reverses even number of pairs, and \(-1\) if \(\sigma\) reverses odd number of pairs.
  • Prove that if \(\sigma\) is a permutation and \(\tau\) is a transposition, then \(\mathrm{isgn}(\sigma\circ\tau)=-\mathrm{isgn}(\sigma)=\mathrm{isgn}(\tau\circ\sigma)\).
  • From the previous point, conclude that \(\mathrm{isgn}(\sigma)=\mathrm{nsgn}(\sigma)\) (thus the sign is well defined).
  • From the definition of \(\mathrm{nsgn}\), show that \(\mathrm{nsgn}(\sigma\circ\tau)=\mathrm{nsgn}(\sigma)\mathrm{nsgn}(\tau)\) for any two permutations \(\sigma\) and \(\tau\).

Task 2. Prove that the number of elements of \(S_n\) (i.e. the order of the symmetric group \(S_n\)) is \(n!\).

7.4 Coursework Sheet 4

Submit a single pdf with scans of your work to Blackboard by Tuesday, 1 March 2022, 17:00.

Exercise 1

The set \(\mathbb{R}^2\) together with the usual vector addition forms an abelian group. For \(a \in \mathbb{R}\) and \(\mathbf{x} = \left(\begin{array}{c}x_1\\x_2\end{array}\right) \in \mathbb{R}^2\) we put \(a \otimes \mathbf{x} := \left(\begin{array}{c} ax_1\\ x_2\end{array}\right) \in \mathbb{R}^2\); this defines a scalar multiplication \[\mathbb{R} \times \mathbb{R}^2 \rightarrow \mathbb{R}^2, \quad (a, \mathbf{x}) \mapsto a \otimes \mathbf{x},\] of the field \(\mathbb{R}\) on \(\mathbb{R}^2\). Determine which of the axioms defining a vector space hold for the abelian group \(\mathbb{R}^2\) with this scalar multiplication. (Proofs or counterexamples are required.)

Exercise 2

The set \(\mathbb{R}_{>0}\) of positive real numbers together with multiplication forms an abelian group. Let \(\mathbb{R}_{>0}^n\) denote the \(n\)-fold cartesian product of \(\mathbb{R}_{>0}\) with itself (cf. Exercise 1 on Sheet 2). (You may find it convenient to use the symbol \(\oplus\) for the binary operation in the abelian group \(\mathbb{R}_{>0}^n\), that is \((b_1, \ldots, b_n) \oplus (c_1, \ldots, c_n) = (b_1 c_1, \ldots, b_n c_n)\) for \(b_1, \ldots, b_n, c_1, \ldots, c_n \in \mathbb{R}_{>0}\).) Furthermore, for \(a \in \mathbb{Q}\) and \(\mathbf{b} = (b_1, \ldots, b_n) \in \mathbb{R}_{>0}^n\) we put \(a \otimes \mathbf{b} := (b_1^a, \dots, b_n^a)\). Show that the abelian group \(\mathbb{R}_{>0}^n\) together with the scalar multiplication \[\mathbb{Q} \times \mathbb{R}_{>0}^n \rightarrow \mathbb{R}_{>0}^n, \quad (a, \mathbf{b}) \mapsto a \otimes \mathbf{b},\] is a vector space over \(\mathbb{Q}\).

Exercise 3

Let \(V\) be a vector space over the field \(F\) and let \(a\in F\) and \(x,y \in V\).

  1. Show that \(a(x-y) = ax-ay\) in \(V\).

  2. If \(ax=0_V\) show that \(a=0_F\) or \(x=0_V\).

(Remember to give a reason for each step.)

Exercise 4

Let \(S\) be a set and let \(V\) be a vector space over a field \(F\). Let \(V^S\) denote the set of all maps from \(S\) to \(V\). We define an addition on \(V^S\) and a scalar multiplication of \(F\) on \(V^S\) as follows: let \(f,g \in V^S\) and let \(a \in F\); then \[(f+g)(s) := f(s) + g(s) \textrm{ and } (af)(s) := a\, (f(s)) \textrm{ (for any } s \in S).\] Show that \(V^S\) is a vector space over \(F\). (For a complete proof many axioms need to be checked. In order to save you some writing, your solution will be considered complete, if you check that there exists an additive identity element in \(V^S\), that every element in \(V^S\) has an additive inverse and that the second distributivity law holds.)

7.5 Coursework Sheet 5

Submit a single pdf with scans of your work to Blackboard by Tuesday, 15 March 2022, 17:00.

Exercise 1

Let \(n \ge 2\). Which of the conditions defining a subspace are satisfied for the following subsets of the vector space \(M_{n\times n}(\mathbb{R})\) of real \((n\times n)\)-matrices? (Proofs or counterexamples are required.) \[\begin{eqnarray*} U& := &\{A \in M_{n \times n}(\mathbb{R})\, | \,\textrm{rank}(A) \le 1\}\\ V& := & \{A \in M_{n \times n}(\mathbb{R})\, | \,\det(A) = 0\}\\ W& := & \{A \in M_{n \times n}(\mathbb{R})\, |\, \textrm{trace}(A) = 0\} \end{eqnarray*}\] (Recall that \(\textrm{rank}(A)\) denotes the number of non-zero rows in a row-echelon form of \(A\) and \(\textrm{trace}(A)\) denotes the sum \(\sum_{i=1}^n a_{ii}\) of the diagonal elements of the matrix \(A = (a_{ij})\).)

Exercise 2

Which of the following subsets of the vector space \(\mathbb{R}^{\mathbb{R}}\) of all functions from \(\mathbb{R}\) to \(\mathbb{R}\) are subspaces? (Proofs or counterexamples are required.) \[\begin{eqnarray*} U&:= & \{f \in \mathbb{R}^{\mathbb{R}}\, | \,f \textrm{ is differentiable and } f'(7)=0\}\\ V&:= & \{f \in \mathbb{R}^\mathbb{R} \,|\, f \textrm{ is polynomial of the form } f = at^2 \textrm{ for some } a \in \mathbb{R}\}\\ &= &\{f \in \mathbb{R}^\mathbb{R} \,|\, \exists a \in \mathbb{R} : \forall s \in \mathbb{R} : f(s) = a s^2\}\\ W&:= & \{f \in \mathbb{R}^\mathbb{R} \,| \,f \textrm{ is polynomial of the form } f = a t^i \textrm{ for some } a \in \mathbb{R}\textrm{ and }i\in\mathbb{N}\}\\ & = &\{f \in \mathbb{R}^\mathbb{R}\, | \,\exists i \in \mathbb{N}\; \exists a \in \mathbb{R} : \forall s \in \mathbb{R} : f(s) = a s^i\}\\ X& := & \{f \in \mathbb{R}^\mathbb{R} \,|\, f \textrm{ is odd}\} \end{eqnarray*}\] (Recall that a function \(f:\mathbb{R} \rightarrow \mathbb{R}\) is called odd if \(f(-s) = -f(s)\) for all \(s \in \mathbb{R}\).)

Exercise 3

Let \(\mathbb{F}_2 = \{0, 1\}\) denote the field with 2 elements.

  1. Let \(V\) be a vector space over \(\mathbb{F}_2\). Show that every non-empty subset \(W\) of \(V\) which is closed under addition is a subspace of \(V\).

  2. Show that \(\{(0,0), (1,1)\}\) is a subspace of the vector space \(\mathbb{F}_2^2\) over \(\mathbb{F}_2\).

  1. Write down all subsets of \(\mathbb{F}_2^2\) and underline those subsets which are subspaces. (No explanations are required.)

Exercise 4 (optional, not marked)

Let \(V\) be a vector space over a field \(F\). Putting \(S=V\) in Example 2.7 we obtain the vector space \(F^V\) consisting of all functions from \(V\) to \(F\). Consider the subset \[V^*:=\{L:V \rightarrow F \;|\; L \textrm{ is a linear transformation}\},\] consisting of all linear transformations from the vector space \(V\) to the (one-dimensional) vector space \(F\). Show that \(V^{*}\) is a subspace of \(F^{V}\). (To get you started, at the end of this sheet you’ll find a detailed proof of the first of the three conditions that need to be verified for a subspace.)

Extra question (not marked, do not submit)

Let \(V\) be a vector space over a field \(F\) and let \(X, Y\) and \(Z\) be subspaces of \(V\), such that \(X\subseteq Y\). Show that \(Y\cap(X+Z) = X+(Y\cap Z)\). (Note: this is an equality of sets, so you need to show that every vector in the LHS also belongs to RHS, and vice versa.)

Verification of the first condition of being a subspace, for \(V^*\) from Exercise 4

(You don’t need to reproduce this in your solution, just say that the first condition is proved.)

The first condition for a subspace asserts that the zero vector of the “big” vector space \(F^V\) belongs to set \(V^*\) that we are showing to be a subspace.

The zero vector ( = the additive identity element for vector addition) of \(F^V\) is the zero function \(\underline{0}:V\to F\), defined by \(\underline{0}(v)=0_F\) for all \(v\in V\), that is, it maps every vector \(v\) from \(V\) to the additive identity element \(0_F\) in the field \(F\).

We need to show that this function \(\underline{0}\) belongs to the set \(V^*\), in other words, that it is a linear transformation from \(V\) to \(F\). This entails checking two conditions:

  1. \(\underline0\) is compatible with addition: take arbitrary vectors \(x,y\in V\). We need to check that \(\underline0(x+y)=\underline0(x)+\underline0(y)\) in \(F\):
    LHS \(= 0_F\) (by definition of \(\underline0\))
    RHS \(= 0_F+0_F = 0_F\) (by definition of \(\underline0\) and the field axioms)
    So LHS = RHS.

  2. \(\underline0\) is compatible with scalar multiplication: take a vector \(x\in V\) and a scalar \(a\in F\). We need to check that \(\underline0(ax)=a(\underline{0}(x))\) in \(F\):
    LHS \(= 0_F\) (by definition of \(\underline{0}\))
    RHS \(= a0_F = 0_F\) (by definition of \(\underline0\) and Prop. 2.3(a))
    So again LHS = RHS.

7.6 Coursework Sheet 6

Submit a single pdf with scans of your work to Blackboard by Tuesday, 22 March 2022, 17:00.

Exercise 1

Which of the following are spanning sets for the vector space \(\mathbb{P}_2\) of polynomial functions of degree at most \(2\)? (Give reasons for your answers.)

  1. \(1, t^2+t, t^2-2\)

  2. \(2, t^2, t, 2t^2 +3\)

  3. \(t+2, t^2+1\)

Exercise 2

Determine whether the following are linearly independent sets of vectors in the vector space \(\mathbb{R}^\mathbb{R}\) of all functions from \(\mathbb{R}\) to \(\mathbb{R}\). (Give reasons for your answers.)

  1. \(1+t\), \(1+t+t^2\), \(1+t+t^2+t^3\), \(1+t+t^2+t^3+t^4\)

  2. \(\sin, \sin^2, \sin^3\)

  3. \(1, \sin^2, \cos^2\)

(Here for example \(\sin^2\) denotes the function \(\mathbb{R} \rightarrow \mathbb{R}, s \mapsto (\sin(s))^2\).)

Exercise 3

Find a basis of the null space \(N(A) \subset \mathbb{R}^5\) of the matrix \[ A = \left(\begin{array}{ccccc} 1 & -2 & 2 & 3 & -1\\ -3 & 6 & -1 & 1 & -7\\ 2 & -4 & 5 & 8 & -4 \end{array}\right) \in M_{3 \times 5}(\mathbb{R}) \] and hence determine its dimension.

Exercise 4

  1. Determine whether the following \((2 \times 2)\)-matrices form a basis of the vector space \(M_{2 \times 2}(\mathbb{R})\) of all \((2 \times 2)\)-matrices over \(\mathbb{R}\):

\[A_1 = \left(\begin{array}{cc} 4 & 0 \\ 0 & 0 \end{array}\right), \quad A_2 = \left(\begin{array}{cc} 3 & 1 \\ 0 & 0 \end{array}\right), \quad A_3 = \left(\begin{array}{cc} 2 & 2 \\ 2 & 0 \end{array}\right), \quad A_4 = \left(\begin{array}{cc} 1 & 3 \\ 1 & 1 \end{array}\right).\]

  1. Find a basis of the subspace \(W:= \{A \in M_{2 \times 2}(\mathbb{R}) \mid \textrm{trace}(A) = 0\}\) of the vector space \(M_{2 \times 2}(\mathbb{R})\) and hence determine the dimension of \(W\). (Recall that \(\textrm{trace}(B)\) of a square matrix \(B=(b_{ij})\in M_{n\times n}(F)\) denotes the sum of its diagonal entries, \(\textrm{trace}(B)=\sum_{i=1}^nb_{ii}\).)

Extra exercise (not marked, do not submit)

We view \(\mathbb{C}^2 = \left\{\left(\begin{array}{c} w \\ z \end{array}\right): w, z \in \mathbb{C}\right\}\) as a vector space over \(\mathbb{C}\), \(\mathbb{R}\) and \(\mathbb{Q}\) (cf. Example 3.16 (b)). Let \({\bf x}_1:= \left(\begin{array}{c} i \\ 0 \end{array}\right)\), \({\bf x}_2:= \left(\begin{array}{c} \sqrt{2} \\ \sqrt{5} \end{array}\right)\), \({\bf x}_3:= \left(\begin{array}{c} 0 \\ 1 \end{array}\right)\), \({\bf x}_4:= \left(\begin{array}{c} i\sqrt{3} \\ \sqrt{3} \end{array}\right)\), \({\bf x}_5:= \left(\begin{array}{c} 1 \\ 3 \end{array}\right) \in \mathbb{C}^2\). Determine \(\textrm{dim}_F(\textrm{Span}_F({\bf x}_1, {\bf x}_2, {\bf x}_3, {\bf x}_4, {\bf x}_5))\) for \(F= \mathbb{C}\), \(\mathbb{R}\) and \(\mathbb{Q}\).

7.7 Coursework Sheet 7

Submit a single pdf with scans of your work to Blackboard by Tuesday, 26 April 2022, 17:00.

Exercise 1

Determine whether the following maps are linear transformations. (For a matrix \(A\), \(A^\mathrm{T}\) denotes its transpose, see Section 2.3 in L.A.I.) (Proofs or counterexamples are required.)

  1. \(L: \mathbb{R}^2 \rightarrow \mathbb{R}^3, \quad \left(\begin{array}{c} x_1 \\ x_2 \end{array}\right) \mapsto \left(\begin{array}{c}x_2\\ 2x_1 - 3x_2 \\ 0\end{array}\right)\) \(\qquad\) (b) \(L: \mathbb{R}^2 \rightarrow \mathbb{R}, \quad \left(\begin{array}{c}x_1 \\ x_2 \end{array}\right) \mapsto x_1^2 + x_2^2\)
  1. \(L: M_{n \times n}(\mathbb{R}) \rightarrow M_{n \times n}(\mathbb{R}), \quad A \mapsto A^\mathrm{T} - A\) \(\qquad\) (d) \(L: \mathbb{P}_3 \rightarrow \mathbb{P}_2, \quad f \mapsto f' + (f(0))t\)

Exercise 2

Consider the linear transformation \(\mathbb{R}^3 \rightarrow \mathbb{R}^5\) given by \(L(\mathbf{x}) = A\mathbf{x}\) where \(A\) is the matrix \[A=\left(\begin{array}{ccc} 1 & 1 & -2\\ 2 & 3 & -3 \\ 3& -4 & -13\\ -1 & 1 & 13 \\ 0 & -8 & 2 \end{array}\right) \in M_{5 \times 3}(\mathbb{R}).\] Find a basis of the image of \(L\). Using the Dimension Theorem show that \(L\) is injective.

Exercise 3

Let \(F\) be a field.

  1. Let \(A \in M_{n \times n}(F)\) be an invertible matrix. Show that the linear transformation \[L_A: F^n \rightarrow F^n, \quad \mathbf{x} \mapsto A\mathbf{x},\] (cf. Example 4.3(a)) is an isomorphism.

  2. Let \(L: V \rightarrow W\) be an isomorphism between vector spaces over \(F\). Show that the inverse map \(L^{-1}:W \rightarrow V\) is a linear transformation (and hence an isomorphism as well).

Exercise 4

For \(\mathbf{y} \in \mathbb{R}^n\) let \(L_{\mathbf{y}}: \mathbb{R}^n \to \mathbb{R}\) denote the map given by \(\mathbf{x} \mapsto L_{\mathbf{y}}(\mathbf{x}) = \mathbf{x} \cdot \mathbf{y}\) where \(\mathbf{x} \cdot \mathbf{y}\) denotes the dot product of \(\mathbf{x}\) and \(\mathbf{y}\) introduced in Linear Algebra I.

  1. For each \(\mathbf{y} \in \mathbb{R}^n\) show that \(L_\mathbf{y}\) is a linear transformation and compute \(\dim_{\mathbb{R}}(\textrm{ker}(L_\mathbf{y}))\).

  2. (optional, not marked) Let \((\mathbb{R}^n)^*\) denote the vector space introduced in Coursework 4/Exercise 4. Show that the map \(L: \mathbb{R}^n \to (\mathbb{R}^n)^*\), \(\mathbf{y} \mapsto L_\mathbf{y}\), is an isomorphism. (Hint: For surjectivity use Proposition 4.4.)

7.8 Coursework Sheet 8

Submit a single pdf with scans of your work to Blackboard by Tuesday, 10 May 2022, 17:00.

Exercise 1

From Calculus we know that for any polynomial function \(f: \mathbb{R} \rightarrow \mathbb{R}\) of degree at most \(n\), the function \(I(f): \mathbb{R} \rightarrow \mathbb{R}\), \(s \mapsto \int_0^s f(u)\, du\), is a polynomial function of degree at most \(n+1\). Show that the map \[I: \mathbb{P}_n \rightarrow \mathbb{P}_{n+1}, \quad f \mapsto I(f),\] is an injective linear transformation, determine a basis of the image of \(I\) and find the matrix \(M\in M_{(n+2)\times(n+1)}(\mathbb{R})\) that represents \(I\) with respect to the basis \(1, t, \ldots, t^n\) of \(\mathbb{P}_n\) and the basis \(1, t, \ldots, t^{n+1}\) of \(\mathbb{P}_{n+1}\).

Exercise 2

  1. Let \(\alpha \in \mathbb{C}\) and \(A:= \left(\begin{array}{ccc} 1-i & \alpha & i \\ i-\alpha & 1 - \alpha & \alpha -i \\ 1- \alpha & 1 & 2+ \alpha \end{array}\right) \in M_{3 \times 3}(\mathbb{C})\). Compute \(\det(A) \in \mathbb{C}\).

  2. Let \(F\) be a field, \(n\) be even and let \(c_1, \ldots, c_n \in F\). Follow the blueprint of the proof of Example 5.2(d) and use Exercise 2(a) on Coursework Sheet 3 to compute the determinant of the matrix \[B:= \left(\begin{array}{cccc} 0 & \ldots & 0 & c_1 \\ \vdots & ⋰ & ⋰ & 0 \\ 0 & ⋰ & ⋰ & \vdots \\ c_n & 0 & \ldots & 0 \end{array}\right) \in M_{n \times n}(F).\]

Exercise 3

Let \(F\) be a field, let \(n\ge 2\) and let \(a_1, \ldots, a_n \in F\). Show that \[\det\left(\begin{array}{cccc} 1 & 1 & \ldots & 1 \\ a_1 & a_2 & \ldots & a_n \\ \vdots & \vdots & & \vdots \\ a_1^{n-1} & a_2^{n-1} & \ldots & a_n^{n-1} \end{array}\right) = \prod_{1 \le i < j \le n} (a_j - a_i);\] i.e. the product is taken over all pairs \((i,j)\) that satisfy \(1 \le i < j \le n\). (Hint: It may help if you do the cases \(n=2\) and \(n=3\) first. In general, use the row operations \(R_n \mapsto R_n - a_1 R_{n-1}, R_{n-1} \mapsto R_{n-1} - a_1 R_{n-2}, \ldots, R_2 \mapsto R_2-a_1 R_1\), then expand along the first column and use Theorem 5.3(b) to obtain a matrix of size \((n-1) \times (n-1)\) which has the same shape as the given matrix. Now use induction on \(n\).)

Exercise 4

Let \(A = \left(\begin{array}{ccc} 1+i & 1-i & 2 \\ 3 & i & -i \\ 1 & 2+i & 2-i \end{array}\right) \in M_{3 \times 3}(\mathbb{C}) \textrm{ and } B = \left(\begin{array}{ccc} 4 & 3+i & 2i \\ 1 & 2-i & 2+2i \\ 1-i & i & 3 \end{array}\right) \in M_{3 \times 3}(\mathbb{C}).\)

Compute \(\det(A)\), \(\det(B)\), \(\det(AB)\) and \(\det(A^2)\).

7.9 Coursework Sheet 9

Submit a single pdf with scans of your work to Blackboard by Tuesday, 17 May 2022, 17:00.

Exercise 1

Let \(F\) be a field and let \(A \in M_{n\times n}(F)\).

  1. If \(n=2\) show that \(p_A(\lambda) = \lambda^2 - \textrm{trace}(A) \lambda + \det(A)\). (Recall that \(\textrm{trace}(B)\) of a square matrix \(B=(b_{ij})\in M_{n\times n}(F)\) denotes the sum of its diagonal entries, \(\textrm{trace}(B)=\sum_{i=1}^nb_{ii}\).)

  2. Let \(k \ge 1\). Show that if \(\lambda\) is an eigenvalue of \(A\) then \(\lambda^k\) is an eigenvalue of \(A^k\).

  3. Suppose that \(F = \mathbb{Q}\), \(\mathbb{R}\) or \(\mathbb{C}\) and that \(A^2 = I_n\). Show that if \(\lambda\) is an eigenvalue of \(A\) then \(\lambda = 1\) or \(\lambda = -1\). Show that \(\textrm{ker}(L_{I_n +A}) = E_{-1}(A)\) and that \(\textrm{im}(L_{I_n+A}) = E_1(A)\). (Note: The notation “\(L_{\mathrm{matrix}}\)” is from Example 4.3 (a).)

Exercise 2

Let \(F\) be a field and let \(A \in M_{n\times n}(F)\) be a diagonalizable matrix.

  1. Let \(k\ge 1\). Show that \(A^k\) is diagonalizable.

  2. Show that the transpose \(A^T\) of \(A\) is diagonalizable.

  3. Show that if \(A\) is invertible then \(A^{-1}\) is diagonalizable.

Exercise 3

Find the eigenvalues of each of the following matrices and determine a basis of the eigenspace for each eigenvalue. Determine which of these matrices are diagonalizable; if so, write down a diagonalizing matrix.

\[A= \left(\begin{array}{ccc} 0 & 0 & -2\\ 1&2&1\\1&0&3 \end{array}\right) \in M_{3 \times 3}(\mathbb{R}), \quad B= \left(\begin{array}{cc}4&1\\-1&2\end{array}\right) \in M_{2 \times 2}(\mathbb{R}),\] \[C=\left(\begin{array}{ccc}1&0&0\\1&-1&2\\1&-1&1\end{array}\right) \textrm{ as element of } M_{3 \times 3}(\mathbb{R}) \textrm{ and as element of } M_{3 \times 3}(\mathbb{C}).\] Compute \(C^{2020}\).

Exercise 4

Let \(V\) be a vector space over a field \(F\) and let \(L, M\) be two linear transformations from \(V\) to itself.

  1. Suppose that \(L \circ M = M \circ L\). Show that \(L(E_\lambda(M)) \subseteq E_\lambda(M)\) for all \(\lambda \in F\).

  2. Suppose that \(V\) is of finite dimension. Show that \(L\) is injective if and only if it is surjective.