Tutorial: Vectors and Linear Algebra

Let \(A = (1,3,-1)\) and \(B = (4,1,1)\), and let \(\mathbf{a}\) and \(\mathbf{b}\) be the position vectors of \(A\) and \(B\) respectively. Determine:

1. the length of \(\mathbf{a}\);
2. the vector having length \(3\) in the same direction as \(\mathbf{a}\);
3. the vector represented by \(\overrightarrow{AB}\);
4. parametric equations for the line through \(A\) and \(B\);
5. the cosine of the angle between \(\mathbf{a}\) and \(\mathbf{b}\);
6. the vector product \(\mathbf{a}\times\mathbf{b}\) of \(\mathbf{a}\) and \(\mathbf{b}\).

load_package avector;
a := avec(1,3,-1);  b := avec(4,1,1);
% (a)
vmod a;
% (b)
3 a/ws;
% (c)
ab := b - a;
% (d)
line := a + alpha*ab;
% Or as scalar equations:
{x = line(0), y = line(1), z = line(2)};
% (e)
(a dot b)/(vmod a * vmod b);
% (f)
a cross b;

Let \(\mathbf{a} = \left(\begin{array}{c}3\\-1\\-6\end{array}\right)\), and let \(\mathbf{b}\) and \(\mathbf{c}\) be position vectors of points in the \((x,y)\)-plane such that the area of the parallelogram with sides \(\mathbf{b}\) and \(\mathbf{c}\) is \(7\). Let the parallelepiped with sides corresponding to \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) have volume \(V\).

1. Is there sufficient information to calculate \(V\)?
2. If so, determine \(V\).

load_package avector;
a := avec(3,-1,-6);
k := avec(0,0,1);
% b cross c = |b cross c| k = 7 k
% V = |a dot (b cross c)| = 7 |a dot k|
% (a) Yes
% (b)
a dot k;  V := 7 abs ws;

Determine all solutions to the following system of linear equations in \(x, y, z\) defined over \(\mathbb{R}\): \[ \left.\begin{eqnarray*} -x-2y+5z &=& 8 \\ 3x+2y+ z &=& -4 \\ x+ y- z &=& -3 \end{eqnarray*}\right\}. \]

eqns := {-x-2y+5z = 8, 3x+2y+z = -4, x+y-z = -3};
solve ws;  % regarding arbcomplex() as a real parameter

Calculate the distance between the parallel lines with vector equations \[ r = \left(\begin{array}{c}2\\1\\2\end{array}\right) + \lambda \left(\begin{array}{c}1\\2\\3\end{array}\right) \quad\mbox{and}\quad r =\left(\begin{array}{c}2\\-1\\3\end{array}\right) + \mu \left(\begin{array}{c}-2\\-4\\-6\end{array}\right) \] respectively. (The parameters \(\lambda\) and \(\mu\) both range over the whole of \(\mathbb{R}\).)

load_package avector;
% This is a vector from one line to the other:
avec(2,1,2) - avec(2,-1,3);
% and this is the magnitude of its projection perpendicular to the lines:
vmod(ws cross avec(1,2,3));

Let \[ A = \left(\begin{array}{rrr}1 & 2 & 0\\-1 & 1 & 2\\1 & -2 & -1\end{array}\right) \quad\mbox{and}\quad B = \left(\begin{array}{rrr}2 & -1 & 0\\-1 & 2 & 0\\1 & -2 & -3\end{array}\right). \] Calculate the following:

1. \(-2A+B\);
2. \(A^2\);
3. the characteristic polynomial of \(B\);
4. \(\det B\) [Hint: the answer to the previous part should help].

a := mat((1,2,0),(-1,1,2),(1,-2,-1));
b := mat((2,-1,0),(-1,2,0),(1,-2,-3));
-2a+b;
a^2;
det(lam*mat((1,0,0),(0,1,0),(0,0,1))-b);
-sub(lam=0,ws);  % using the hint, or directly:
det b;

Let \(S_\theta\) denote the \(2\times2\) matrix representing the reflexion (in the \((x,y)\)-plane) in the line through the origin at anticlockwise angle \(\theta/2\) to the \(x\)-axis. Then \[ S_\theta = \left(\begin{array}{cc} \cos\theta & \sin\theta \\ \sin\theta\ & -\cos\theta \end{array}\right). \]

1. Determine the eigenvalues of \(S_\theta\).
2. Now let \(S\) be the matrix of the reflexion in the line \(y = x\). Write down \(S\), and write down an eigenvector of \(S\) and its corresponding eigenvalue.

S_theta := mat((cos theta, sin theta),(sin theta, -cos theta));
% (a)
let cos(~th)^2 + sin(~th)^2 => 1;
E_theta := mateigen(S_theta, lam);  % Hence eigenvalues are +/- 1.
% (b)
theta := pi/2;
S := S_theta;
E_theta;
third first ws; % with corresponding eigenvalue 1.

Compute the determinant of the matrix \[ \left(\begin{array}{ccc} 1 & -1 & 1 \\ 2 & 0 & 1 \\ -1 & 2 & -2 \end{array}\right). \] Is this matrix invertible? If so, find its inverse. Hence or otherwise solve the linear system \[ \begin{eqnarray} x_1 - x_2 + x_3 &=& 1 \\ 2x_1 + x_3 &=& 2 \\ -x_1 + 2x_2 - 2x_3 &=& -3. \end{eqnarray} \]

M := mat((1,-1,1),(2,0,1),(-1,2,-2));
det M;
M^-1;
% Hence:
ws * mat((1),(2),(-3));
% or otherwise:
eqns := {x1 - x2 + x3 = 1, 2x1 + x3 = 2, -x1 + 2x2 - 2x3 = -3};
solve ws;

Let \[ A = \left(\begin{array}{cc} 0 & 3 \\ -1 & 2 \end{array}\right), \quad B =\left(\begin{array}{cc} -1 & 2 \\ 0 & 3 \\ 1 & -2 \end{array}\right). \] For each of the products \(A^2\), \(AB\), \(BA\), \(B^2\), state whether or not it exists; if it exists then evaluate it.

A := mat((0,3),(-1,2));  B := mat((-1,2),(0,3),(1,-2));
A^2;  A*B;  B*A;  B^2;

Suppose that \(B\) is a square matrix and \(B^2 = O\). Prove that \(I+B\) is invertible, and find its inverse. Illustrate with a \(2\times2\) example.

% (I + B)(I - B) = I^2 - B^2 = I, hence (I + B)^-1 = (I - B)
B := mat((0,1),(0,0));  B^2;
Id := mat((1,0),(0,1));  A := Id + B;
A^-1 = Id - B;

For each of the following statements about \(n\times n\) matrices \(A\) and \(B\), state whether it is true or false, and illustrate with general \(2\times2\) matrices.

1. \(\det(A + B) = \det A + \det B\);
2. \(\det(A + B^T) = \det(B + A^T)\);
3. \(\det(BA) = \det(AB)\);
4. \(\det(\lambda A) = \lambda \det A\), where \(\lambda\in\mathbb{R}\).

matrix A, B;  operator aa, bb;
for i := 1:2 do for j := 1:2 do <<A(i,j) := aa(i,j); B(i,j) := bb(i,j)>>;
A;  B;
% (a) false
det(A + B) = det A + det B;
if det(A + B) = det A + det B then true else false;
% (b) true
det(A + tp B)= det(tp A + B);
if det(A + tp B)= det(tp A + B) then true else false;
% (c) true
det(B*A) = det(A*B);
if det(B*A) = det(A*B) then true else false;
% (d) false (unless n=1)
det(lam*A) = lam*det(A);
if det(lam*A) = lam*det(A) then true else false;

Let \(V = \mathbb{R}^4\), and let \[ S = \{(0,2,1,-3)^T, (1,1,1,2)^T, (3,0,-1,-1)^T, (-1,4,4,2)^T\}. \] Either determine whether or not \(S\) is a spanning set for \(V\), or determine whether or not \(S\) is a linearly dependent set.

% Using lists to represent vectors:
S := {{0,2,1,-3}, {1,1,1,2}, {3,0,-1,-1}, {-1,4,4,2}};
rank S;
% or equivalently
S := mat((0,2,1,-3), (1,1,1,2), (3,0,-1,-1), (-1,4,4,2));
rank S;
% S is not a spanning set because it contains only 3 linearly independent vectors.

Let \(P_2\) denote the set of polynomials of degree at most \(2\), that is \[ P_2 = \{\mathbf{p} \mid \mathbf{p}(x) = a_2x^2 + a_1x + a_0 \mbox{ for some } a_0, a_1, a_2 \in \mathbb{R}\}. \] Let \(D : P_2 \to P_2\) be the mapping given by \(D(\mathbf{p}) = \mathbf{q}\), where \(\mathbf{p}'\) denotes the derivative of \(\mathbf{p}\), and \[ \mathbf{q}(x) = x^2\mathbf{p}(0) + \mathbf{p}'(x). \]

1. Show that \(D\) is a linear transformation.
2. Is \(D\) surjective? Justify your answer.
3. Is \(D\) injective? Justify your answer.

operator D;  let D(~p) => x^2*sub(x=0,p) + df(p,x);
p1 := a2*x^2 + a1*x + a0;  p2 := b2*x^2 + b1*x + b0;
% (a)
D(alpha*p1 + beta*p2) = alpha*D(p1) + beta*D(p2);
if D(alpha*p1 + beta*p2) = alpha*D(p1) + beta*D(p2) then true else false;
% (b) Yes if we can find p1 that maps to an arbitrary p2:
D(p1) = p2;
coeff(lhs ws - rhs ws, x);
solve(ws, {a2,a1,a0});
% (c) Yes, since precisely one p1 maps to each p2.

Let \(H\) be a subspace of \(\mathbb{R}^n\) and let \(H^\perp\) denote the orthogonal complement of \(H\). Suppose \(n = 4\) and \[ H = \{(x,y,z,t)^T \mid x + 2y - 3z = 0, 3y + 2z - t = 0\}. \] Compute \(\dim H\), \(\dim H^\perp\), and a basis for \(H^\perp\).

% First, ensure that arbitrary constants have the expected indices:
% (See MacCallum and Wright, footnote 31 on page 35.)
symbolic(!!arbint := 0);
solve({x+2y-3z=0, 3y+2z-t=0}, {x,y,z,t});
h := for each el in first ws collect rhs el;
% 2 parameters, so dim H = 2, with basis vectors
h1 := sub(arbcomplex(1)=1, arbcomplex(2)=0, h);
h2 := sub(arbcomplex(1)=0, arbcomplex(2)=1, h);
% Hence a vector {x,y,z,t} in Hperp must satisfy
solve({-2x+1y+0z+3t=0, 3x+0y+1z+2t=0}, {x,y,z,t});
hp := for each el in first ws collect rhs el;
% 2 parameters, so dim Hperp = 2, with basis vectors
hp1 := sub(arbcomplex(3)=1, arbcomplex(4)=0, hp);
hp2 := sub(arbcomplex(3)=0, arbcomplex(4)=1, hp);

Let \[ A = \left(\begin{array}{ccc} -1 & 0 & 1 \\ 2 & 3 & 2 \\ 2 & 4 & 0 \end{array}\right), \quad \mathbf{v}_1 = \left(\begin{array}{c} -2 \\ 1 \\ 0 \end{array}\right), \quad \mathbf{v}_2 = \left(\begin{array}{c} 1 \\ 0 \\ -1 \end{array}\right). \]

1. Show that \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are eigenvectors of \(A\) and find the corresponding eigenvalues.
2. Find the characteristic polynomial of \(A\).
3. Determine all eigenvalues of \(A\) and find bases for the corresponding eigenspaces.
4. Find an invertible matrix \(P\) and a diagonal matrix \(D\) such that \(P^{-1}AP = D\).

A := mat((-1,0,1),(2,3,2),(2,4,0));
v1 := mat((-2),(1),(0));
v2 := mat((1),(0),(-1));
% (a)
A*v1;  A*v2;
% Hence the eigenvalues are respectively -1 and -2.
% (b)
det(lam*mat((1,0,0),(0,1,0),(0,0,1))-A);
% (c)
solve ws;  ev := mateigen(A, lam);
for each el in ev collect third el;
bases := (ws where arbcomplex(~j) => 1);
% (d)
% Construct D from the eigenvalues:
for each el in ev collect rhs first solve first el;
clear D;  matrix D(3,3);
for i := 1:3 do D(i,i) := part(ws, i);  D;
% Construct P from the eigenspace bases:
matrix P(3,3);
for i := 1:3 do for j := 1:3 do P(i,j) := <<v:=part(bases, j); v(i,1)>>;
% Check
P^(-1)*A*P = D;