Exercises

Exercise 7.1

(See Solution 7.1.) Let \(V = P_{\le 2} = \{ at^2 + bt+c \ | \ a,b,c \in {\bf R} \}\) be the vector space of (real) polynomials of degree \(\le 2\). We consider the scalar product in Example 7.16 3., i.e.,

\[ {\left \langle p, q \right \rangle} = \int_{-1}^1 p(x)q(x) dx. \]

Let \(e_1 = 1\), \(e_2 = t\) and \(e_3 = t^2\). (These vectors form a basis of \(P_{\le 2}\).) Compute \({\left \langle e_i, e_j \right \rangle}\) for \(1 \le i,j \le 3\). (This requires knowledge of basic integration techniques.)
Apply the Gram–Schmidt orthogonalization procedure to this basis.

Exercise 7.2

(See Solution 7.3.) Consider the subspace \(U \subset {\bf R}^3\) given by the solutions of the homogeneous linear system

\[ x-y+3z=0. \]

Find a basis of \(U\).
Compute a basis of \(U^\bot\). What is \(\dim U^\bot\)?
Consider \(t=(0,1,5)\). Find its orthogonal projection onto \(U\) (recall from Corollary 7.34 that \(t = t_U + t_\bot\) with uniquely determined vectors \(t_U \in U\) and \(t_\bot \in U^\bot\). The orthogonal projection of \(t\) onto \(U\) is then the vector \(t_U\).)

Exercise 7.3

(See Solution 7.2.) Consider the subspace \(W \subset {\bf R}^4\) given by the equations

\[ \begin{align*} x-t & = 0\\ y+z-t & = 0\\ \end{align*} \]

(where \(x,y,z,t\) are the coordinates of \({\bf R}^4\)).

Compute a basis of \(W\) and of \(W^\bot\).
Compute the orthogonal projection of \(t=(1,5,1,6)\) onto \(W\).

Exercise 7.4

(See Solution 7.4.) Consider the subspace \(U \subset {\bf R}^3\) given by the equations

\[ \begin{align*} x & = 0\\ x+y+z & = 0\\ \end{align*} \]

(where \(x,y,z\) are the coordinates of \({\bf R}^3\)).

Compute a basis of \(U\) and of \(U^\bot\).
Compute the orthogonal projection of \(t=(5,1,3)\) onto \(U\).

Exercise 7.5

(See Solution 7.5.) Compute the orthogonal complement of \(T = L((1,0,-3))\).

Exercise 7.6

(See Solution 7.6.) Is there a subspace \(U \subset {\bf R}^3\) such that

the orthogonal projection of \(t=(1,1,0)\) onto \(U\) is given by \((1,5,6)\)?
the orthogonal projection of \(t=(2,0,1)\) onto \(U\) is given by \((1,1,1)\)?

Exercise 7.7

(See Solution 7.7.) Let \(L = \left ( \begin{array}{c} 1 \\ 3 \\ 5 \end{array} \right ) + L (\left ( \begin{array}{c} 1 \\ 1 \\ 4 \end{array} \right ))\). Compute the closest point of \(L\) to the origin, and its distance to the origin.

Exercise 7.8

(See Solution 7.8.) Consider the two lines \(L: x=1+t,y=t,z=2+t, t \in {\bf R}\) and \(L':x-3=y-1=z-3\). Are they parallel? Compute the distance between \(L\) and \(L'\).

Exercise 7.9

(See Solution 7.9.) Are the lines

\[ L: x = y-1=-z \text{ and } L' :x-2=-y=\frac z2 \]

identical, parallel, or skew? Compute their distance.

Exercise 7.10

(See Solution 7.10.) Let \(P\) be the plane given by the equation

\[ 4x+5y+10z-20 = 0. \]

Let \(L\) be the line given by the equations \(x = 0, y = 5 - z\).

Sketch \(P\) and \(L\).
Compute the orthogonal complement of the underlying vector space \(W\) of \(P\).
Compute the point of \(P\) that is closest to the origin and its distance to the origin.
Are \(P\) and \(L\) parallel?

Exercise 7.11

(See Solution 7.11.) Which of the following matrices is orthogonally diagonalizable? If so, find a orthonormal eigenbasis of \({\bf R}^2\).

\(A = \left ( \begin{array}{cc} 1 & 2 \\ 2 & 1 \end{array} \right )\)
\(A = \left ( \begin{array}{cc} 1 & 2 \\ -2 & 1 \end{array} \right )\)
\(A = \left ( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right )\).

Exercise 7.12

(See Solution 7.12.) Let \(A\) be a symmetric matrix and \(\lambda \ne \mu\) two distinct eigenvalues of \(A\), with eigenvectors \(v\) and \(w\), respectively. Then \(v \bot w\), i.e., eigenvectors of distinct eigenvalues are orthogonal.

Exercise 7.13

(See Solution 7.13.) Let \(P \subset {\bf R}^4\) by the hyperplane given by

\[ 2 x_1 + x_3 - x_4 = 4, \]

where \((x_1, \dots, x_4)\) are the coordinates of \({\bf R}^4\). For a parameter \(t \in {\bf R}\), let \(L_t\) be the line

\[ L_t = (1,0,0,-2t) + L(t, 1, 0, -1). \]

For which \(t \in {\bf R}\) is \(L_t\) parallel to \(P\)?
Let now \(t = -\frac 12\) and consider the line \(L = L_{-\frac 12}\). Determine the pair(s) of points \((p, l)\) such that \(p \in P\) and \(l \in L\) such that their distance is minimal.

Exercise 7.14

(See Solution 7.14.) Let \(L \subset {\bf R}^3\) be the line defined be the system

\[ \begin{align*} x+z & = 1\\ y+z & = -2. \end{align*} \]

Let \(L'\) be the line in \({\bf R}^3\) passing through the points \((0,0,1)\) and \((0,1,1)\).

Present \(L\) as \(L=v+W\) for a subspace \(W \subset {\bf R}^3\). Do the same for \(L'\).
Are \(L\) and \(L'\) a) identical, b) parallel, c) skew or d) intersecting?
Compute the Cartseian equation (i.e., in the form \(ax+by+cz=d\), for appropriate values of \(a, \dots, d\)) of the plane \(P \subset {\bf R}^3\) that contains \(L\) and is parallel to \(L'\).
Let \(l = (2,-1,-1) \in L\). Compute a point \(l' \in L'\) such that the line passing through \(l\) and \(l'\) is parallel to the plane given by the equation \(x+z-1=0\).

Exercise 7.15

(See Solution 7.15.) Let \(V = P_{\le 2}\) be the vector space of polynomials of degree \(\le 2\). We write elements of \(V\) as \(p(t)=a+bt+ct^2\), where \(a,b,c\in{\bf R}\). Define

\[ {\left \langle p, q \right \rangle} := \int_{-1}^1 p(t) q(t) dt. \]

Confirm that \({\left \langle -, - \right \rangle}\) is a scalar product on \(V\).
Compute an orthonormal basis of \(V\).
Consider the map

\[ f : V \to V, f(p) := t \frac{\partial p}{\partial t} \]

(i.e., it maps a polynomial \(p\) to the product of the indeterminate \(t\) with the derivative of \(p\) with respect to the variable \(t\)). Confirm that this map is linear. Compute the matrix of \(f\) with respect to the standard basis \(e_1 = 1\), \(e_2 = t\) and \(e_3 = t^2\). Is this basis an eigenbasis for \(f\)? Compute \(\dim \ker f\) and \(\dim {\operatorname{im}\ } f\).

Does the map \(f\) have an orthonormal eigenbasis?

Exercise 7.16

(See Solution 7.16.) Consider the subspace \(U \subset{\bf R}^4\) given by the solutions of the equation

\[ x_1-x_2+x_3+2x_4 = 0. \]

(As usual \(x_1, \dots, x_4\) are the coordinates of \({\bf R}^4\).)

Find a basis of \(U\). What is \(\dim U\)?
Compute an orthonormal basis of \(U\).
Compute the orthogonal projection of \(v=(2,3,0,0)\) and of \(w=(2,5,3,0)\) onto \(U\).
Compute \(U^\bot\).

Exercise 7.17

(See Solution 7.17.) In the Euclidean space \({\bf R}^4\), endowed with the standard scalar product, let \(U\) be the subspace spanned by the vectors \(u_1 = (1,2,0,-1)\), \(u_2 = (0,-4,3,4)\).

Compute an orthogonal basis of \(U\).
Compute a basis of \(U^\perp\).
Compute the orthogonal projection of \(v = (0,5,3,4)\) onto \(U\).
Let \(w = (2,-1,0,2)\). Decide whether there is a subspace \(L \subset {\bf R}^4\) such that the orthogonal projection of \(w\) onto \(L\) is the vector \(\ell = (1,1,2,0)\).

Exercise 7.18

(See Solution 7.18.) Consider the following two lines in \({\bf R}^3\), where \(x, y, z\) are the coordinates:

\[ L: \left\{\begin{aligned} & x + y - 1 = 0\\ & 2x - z - 1 = 0 \end{aligned}\right. \qquad M: \left\{\begin{aligned} & x - 2y - 1 = 0\\ & y - z + 2 = 0 \end{aligned}\right. \]

Determine whether \(L\) and \(M\) are the same line, parallel, or skew.
Compute the cartesian equation of the plane that contains the line \(M\) and that is parallel to \(L\). (Recall that a cartesian equation is of the form \(\langle x, a \rangle = d\) for an appropriate vector \(a\) and an appropriate \(d \in {\bf R}\).)
Given the point \(l = (0,1,-1) \in L\) compute a point \(m \in M\) such that the line passing through \(l\) and \(m\) is parallel to the plane defined by the equation \(3x - z = 0\).
Consider the family of planes \(\pi_\alpha: z = \alpha\), for some parameter \(\alpha \in {\bf R}\). Let \(r_\alpha = L \cap \pi_\alpha\) and \(s_\alpha = M \cap \pi_\alpha\). Let \(m_\alpha\) be the midpoint of the segment with endpoints \(r_\alpha\) and \(s_\alpha\). Verify that the points \(m_\alpha\) are all lying on the same line. Moreover, determine the parametric equation of that line.

Exercise 7.19

(See Solution 7.19.) Consider the points \(p = (3,1,0)\), \(q = (0,1,3)\) and \(r = (-3,0,-3) \in {\bf R}^3\). Let \(L\) be the line passing through \(p\) and \(q\).

Determine the parametric equation of \(L\), i.e., express \(L\) in the form \(L = v + W\), for an appropriate vector \(v \in {\bf R}^3\) and a subspace \(W \subset {\bf R}^3\).
Verify that \(r\) does not lie on \(L\). Give the plane \(P\) containing \(p, q, r\) both in vector and in cartesian form.

Exercise 7.20

(See Solution 7.20.) Consider the line \(L = (3,1,0) + L(1,0,-1)\). Is there a plane containing \(L\) and the line \(M\) given by the systen \(x + z = 2\), \(x - 2 y = 2\) (with \(x, y, z\) being the coordinates of \({\bf R}^3\))?

Exercise 7.21

(See Solution 7.21.) Consider the line \(L = (3,1,0) + L(1,0,-1)\). Let \(p = (-1,-1,-1)\). Describe all the points \(q \in {\bf R}^3\) such that the line \(M\) passing through \(p\) and \(q\) intersects \(L\) orthogonally (i.e., intersects it, and does so orthogonally).

Exercise 7.22

(See Solution 7.22.) Consider the plane \(P = \{x \in {\bf R}^3, 3x - 4y+z = 2\}\) and the point \(p = (0,1,6) \in P\), as well as the line \(L = (0,0,2) + L(1, 1, 1) \subset P\), compute the line \(M \subset P\) that is orthogonal to \(L\) and contains the point \(p\).

Exercise 7.23

(See Solution 7.23.) Consider the lines (in \({\bf R}^3\))

\[ \begin{align*} L_1 : \ & 2x-y=-3 \\ & y+z=-2 \end{align*} \]

and

\[ \begin{align*} L_2 : \ & x=t \\ & y = 2 \\ & z = 4-t, t \in {\bf R}. \end{align*} \]

Determine their relative position (skew, parallel etc.)
Find the plane \(\pi\) parallel to \(L_1\) and containing \(L_2\).
Compute the distance between \(\pi\) and \(L_1\).
Find points \(p_1 \in L_1\), \(p_2 \in L_2\) such that

\[ |\hspace{-0.5mm}| {p_1-p_2} |\hspace{-0.5mm}| = d(p_1, p_2) = d(L_1, L_2), \]

i.e., two points that realize the minimal distance between \(L_1\) and \(L_2\).

Exercise 7.24

(See Solution 7.24.) We endow \({\bf R}^4\) with its usual scalar product. We consider the subspace \(U \subset {\bf R}^4\) defined by the equations

\[ \left\{\begin{aligned} & x_1 + x_3 = 0 \\ & 2x_1 + x_2 - x_4 = 0 \end{aligned}\right. \]

Compute an orthogonal basis of \(U\).
Given the vector \(w_1 = (1,1,-1,-1)\) find a vector \(w_2\) that is orthogonal to \(w_1\) and such that the vector space \(W := L(w_1, w_2)\) satisfies \(W = U^\perp\).
Write down a system of linear equations in the unknowns \(x_1, x_2, x_3, x_4\) whose solution set is the subspace \(W = U^\perp\).
Given the vector \(v = (3,1,-1,1)\) find a vector \(u \in U\) such that the vector \(v+u\) has minimal norm.

Exercise 7.25

(See Solution 7.25.) Consider the lines (contained in \({\bf R}^3\))

\[ L: \left\{\begin{aligned} & x - 2y + 4 = 0\\ & y + z -3 = 0 \end{aligned}\right. \qquad M: \left\{\begin{aligned} & x+2z-5 = 0\\ & x-2y + 1 = 0 \end{aligned}\right. \]

Verify that \(L\) and \(M\) are parallel and write down the cartesian equation of the plane containing \(L\) and \(M\).
Given the point \(p = (0,2,1) \in L\) find the point \(q \in M\) such that the line passing through \(p\) and \(q\) is orthogonal to \(L\) and to \(M\).
Write down the cartesian equation of the plane \(X\) containing the line \(L\) and passing through the point \(r = (-1,1,0)\).
Write down the parametric equation of the line \(N\) contained in the plane \(X\), passing through \(r = (-1,1,0)\) and orthogonal to the line \(L\).

Exercise 7.26

(See Solution 7.26.) Let \(U \subset {\bf R}^4\) be the subspace defined by the equations

\[ U: \left\{\begin{aligned} & x_1 - 2x_2 + x_4 = 0 \\ & 3x_2 + x_3 - 2x_4 = 0 \\ & 3x_1 + 2x_3 + t\,x_4 = 0 \end{aligned}\right. \]

Find the value of \(t\) for which \(U\) has dimension 2. For this value of \(t\), compute a basis of \(U\).
Apply the Gram–Schmidt method to the basis computed in part (a) in order to compute an orthogonal basis of \(U\).
For the value of \(t\) computed in part (a), compute a basis of \(U^\perp\).
Given \(v = (3,2,2,-2) \in {\bf R}^4\) compute the cartesian equation of a subspace \(W\) of dimension \(3\) such that the orthogonal projection of \(v\) onto \(W\) is equal to the vector \(w = (1,2,1,1)\).

Exercise 7.27

(See Solution 7.27.) We consider the following points (in \({\bf R}^3\)) \(A = (6,-1,-4)\), \(B = (1,1,-1)\) and the plane \(X: 2x - y - 2z = 3\).

Verify that \(B \in X\). Let \(C\) be the orthogonal projection of \(A\) onto the plane \(X\). Compute \(C\).
Compute the cartesian equation of the plane containing the triangle \(\triangle ABC\).
Compute the parametric equation of the line that a) passes through \(B\), b) is contained in the plane \(X\) and c) is orthogonal to the line passing through \(A\) and \(B\).
Compute the value of the parameter \(t\) such that the line \(M_t: \left\{\begin{aligned} & t\,x - y + 2 = 0 \\ & x + z + 1 = 0 \end{aligned}\right.\) is parallel to the plane \(X\).