Exercises

Exercise 4.1

(See Solution 4.1.) Determine which \(2 \times 2\)-matrix \(A\) is such that the function

\[ f : {\bf R}^2 \to {\bf R}^2, v \mapsto Av \]

are the following:

\(f(v)\) is the point \(v\) reflected along the \(y\)-axis,
\(f(v)\) is the same point as \(v\),
\(f(v)\) is the origin \((0,0)\),
\(f(v)\) is the point \(v\) reflected along the line \(\{(x,x) \ | x \in {\bf R}\}\) (i.e., the “southwest-northeast diagonal”),
\(f(v)\) is the point \(v\) rotated counterclockwise, resp. clockwise by \(60^\circ\)?

Exercise 4.2

(See Solution 4.2.) Determine the matrix \(A\) such that \(Av = \left ( \begin{array}{c} -y \\ x \end{array} \right )\). Describe the behaviour of the function \(v \mapsto Av\) geometrically.

Exercise 4.3

(See Solution 4.3.) Write down the matrix \(A\) such that the function \(f : {\bf R}^4 \to {\bf R}^3, v \mapsto Av\) satisfies

\[ f((1,0,0,0)) = (1,2,3), f((0,1,0,0)) = (0, 0, 7), \]

\[ f((0,0,1,0)) = (0, 0, 0), f((0,0,0,1)) = (13, 0, -1). \]

Determine \(\ker f\) and \({\operatorname{im}\ } f\) (i.e., determine a basis and their dimension).

Exercise 4.4

(See Solution 4.4.) Compute the rank of

\[ A = \left ( \begin{array}{cccc} 0 & 1 & 2 & 1 \\ 1 & 1 & 1 & 0 \\ 0 & -1 & 1 & 1 \\ 1 & 1 & 4 & 2 \end{array} \right ). \]

Exercise 4.5

(See Solution 4.5.) Consider the linear map \(f : {\bf R}^3 \to {\bf R}^3\) described in Example 4.43. Determine the matrix of \(f\) with respect to the standard basis \(e_1, e_2, e_3\) (both in the “source” \({\bf R}^3\), and also in the “target” \({\bf R}^3\)).

Exercise 4.6

(See Solution 4.6.) For \(\lambda \in {\bf R}\) consider the subspace of \({\bf R}^3\) defined as

\[ W_\lambda = L((1,1+\lambda,-1), (2,\lambda-2,\lambda+2)). \]

For each \(\lambda \in {\bf R}\), find a basis and the dimension of \(W_\lambda\).

Exercise 4.7

(See Solution 4.7.) Determine the rank of

\[ \left ( \begin{array}{ccc} \alpha & 0 & 0 \\ 0 & \alpha & 1+\alpha \\ \alpha & 1 & 2 \end{array} \right ) \]

for each \(\alpha \in {\bf R}\).

Exercise 4.8

(See Solution 4.9.) Consider the linear map

\[ f : {\bf R}^2 \to {\bf R}^3, \left ( \begin{array}{c} x_1 \\ x_2 \end{array} \right ) \mapsto \left ( \begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \end{array} \right ) \left ( \begin{array}{c} x_1 \\ x_2 \end{array} \right ) = \left ( \begin{array}{c} x_1+2x_2 \\ x_2 \\ 3x_1+5x_2 \end{array} \right ). \]

Determine \(\ker f\).
Does the vector \(\left ( \begin{array}{c} 1 \\ 0 \\ 3 \end{array} \right )\) lie in the image of \(f\)?

Exercise 4.9

(See Solution 4.10.) Consider the linear map

\[ f : {\bf R}^4 \to {\bf R}^3 \]

\[ \left ( \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array} \right ) \mapsto \left ( \begin{array}{cccc} 2 & -1 & 1 & 1 \\ 0 & 5 & -3 & -5 \\ 3 & -4 & 3 & 4 \end{array} \right ) \left ( \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array} \right ) = \left ( \begin{array}{c} 2x_1-x_2+x_3+x_4 \\ 5x_2-3x_3-5x_4 \\ 3x_1-4x_2+3x_3+4x_4 \end{array} \right ). \]

Determine \(\ker f\).
Determine \(f^{-1}(\left ( \begin{array}{c} 1 \\ -3 \\ -3 \end{array} \right ))\), i.e., find all the vectors \(v \in {\bf R}^4\) such that \(f(v) = \left ( \begin{array}{c} 1 \\ -3 \\ -3 \end{array} \right )\). Is this subset of \({\bf R}^4\) a subspace?

Exercise 4.10

(See Solution 4.11.) Consider the matrix

\[ A_t = \left ( \begin{array}{cccc} 1 & 3 & -1 & 2 \\ 1 & 5 & 1 & 1 \\ 2 & 4 & t & 5 \end{array} \right ). \]

Here \(t \in {\bf R}\) is an arbitrary real number.

Determine the rank of \(A_t\).
Set \(t = -4\). For which \(\alpha \in \bf R\) does the system

\[ A_{-4} x = \left ( \begin{array}{c} 1 \\ \alpha \\ 0 \end{array} \right ) \]

have solutions?

Set again \(t = -4\). Determine the solutions of the system

\[ A_{-4}x = \left ( \begin{array}{c} 1 \\ 4 \\ -1 \end{array} \right ). \]

Is there any \(t \in {\bf R}\) such that the homogeneous system

\[ A_t x = 0 \]

has only the trivial solution (i.e., only the zero vector)?

Exercise 4.11

(See Solution 4.12.) Let \(f : V \to W\) be a linear map. For a subspace \(U \subset V\) we define the image of \(U\) to be

\[ f(U) := \{ f(v) \ | \ v \in U\}. \]

(For example, for \(U = V\), this gives back the image of \(f\) as defined in Definition 4.20).

Arguing as in Proposition 4.23, prove that \(f(U)\) is a subspace of \(W\).
Prove that \(\dim f(U) \le \dim U\).

Exercise 4.12

(See Solution 4.13.) Let \(f : V \to W\) be a linear map. For a subspace \(U \subset W\), we define the preimage of \(U\) to be

\[ f^{-1}(U) := \{ v \in V \ | \ f(v) \in U\}. \]

(For example, if \(U = \{0_W\}\) is the subspace consisting only of the zero vector of \(W\), this gives back the kernel: \(\ker f = f^{-1}(\{0_W\})\).)

Arguing as in Proposition 4.23, prove that \(f^{-1}(U)\) is a subspace of \(V\).

Exercise 4.13

(See Solution 4.14.) Consider the linear map \(f : {\bf R}^4 \to {\bf R}^4\) given by multiplication with the matrix

\[ A = \left ( \begin{array}{cccc} 2 & -1 & -\frac 52 & 1 \\ -1 & 0 & 1 & -\frac12 \\ 1 & 1 & -\frac 12 & \frac 12 \\ 0 & 2 & 1 & 0 \end{array} \right ). \]

Determine \(\ker f\), \({\operatorname{im}\ } f\) and \(\ker f \cap {\operatorname{im}\ } f\).

Exercise 4.14

(See Solution 4.15.) Consider the linear map

\[ f : {\bf R}^3 \to {\bf R}^2 \]

given by

\[ f(x,y,z) = (2x-z, x+y+z). \]

Determine the matrix of \(f\) with respect to the standard basis in \({\bf R}^3\) and the standard basis \({\bf R}^2\).
Determine \(\ker f\) and \({\operatorname{im}\ } f\).
Determine the preimage \(f^{-1}((0,1))\). Write down the linear system whose solution set is this preimage. Is it a subspace of \({\bf R}^3\)?
Show that the vectors \(v_1 = (0,1,2)\), \(v_2 = (0,-1,1)\) and \(v_3 = (1,1,1)\) are a basis of \({\bf R}^3\). Determine the matrix of \(f\) with respect to this basis of \({\bf R}^3\) and the standard basis in the codomain \({\bf R}^2\).

Exercise 4.15

(See Solution 4.16.) Consider the linear map \(f: {\bf R}^3 \to {\bf R}^3\) whose matrix with respect to the standard basis (of both the domain and the codomain \({\bf R}^3\)) is

\[ \left ( \begin{array}{ccc} 2 & -1 & 0 \\ 1 & 0 & 2 \\ 0 & 2 & -1 \end{array} \right ). \]

Let \(v_1 = (1,1,0)\). Compute \(v_2 = f(v_1)\) and \(v_3 = f(v_2)\). Show that \(v_1, v_2, v_3\) form a basis of \({\bf R}^3\).
Consider \(v_4 = f(v_3)\) and determine \(a_1, a_2, a_3\) such that

\[ v_4 = a_1 v_1 + a_2 v_2 + a_3 v_3. \]

Determine the matrix of \(f\) with respect to the basis \(v_1, v_2, v_3\) (both of the domain and of the codomain \({\bf R}^3\)).

Exercise 4.16

(See Solution 4.17.) Consider the linear map

\[ \begin{alignat*} {3} f: {\bf R}^4 &\rightarrow {\bf R}^3 \\ \begin{pmatrix} x\\ y\\ z\\ t\end{pmatrix} &\mapsto \begin{pmatrix} -x+z\\ -y+t\\ x-y \end{pmatrix} \end{alignat*} \]

Write the matrix associated to \(f\) with respect to the standard basis of the domain \({\bf R}^4\) and the standard basis of the codomain \({\bf R}^3\).
Determine \(\ker f\) and \({\operatorname{im}\ } f\).

Exercise 4.17

(See Solution 4.18.) Consider the following functions:

\[ \begin{align*} f_1 & : {\bf R}^2 \rightarrow {\bf R}^3, \left ( \begin{array}{c} x_1 \\ x_2 \end{array} \right ) \mapsto \left ( \begin{array}{c} x_2 + x_1 \\ 3 x_1 \\ 2 x_2 \end{array} \right ) \\ f_2 & : {\bf R}^3 \rightarrow {\bf R}^2, \left ( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right ) \mapsto \left ( \begin{array}{c} x_3 + x_1 \\ 3 x_2 + 4x_1 + 1 \end{array} \right ) \\ f_3 & : {\bf R}^3 \rightarrow {\bf R}^2, \left ( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right ) \mapsto \left ( \begin{array}{c} x_3^2 \\ x_2+x_1 \end{array} \right ) \\ f_4 & : {\bf R}^3 \rightarrow {\bf R}^3, \left ( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right ) \mapsto \left ( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right ) \\ f_5 & : {\bf R}^3 \rightarrow {\bf R}^3, \left ( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right ) \mapsto \left ( \begin{array}{c} x_1+x_2 \\ x_2+x_3 \\ x_3+x_1 \end{array} \right ) \end{align*} \]

Verify which ones are linear.
Determine their images.
For the linear maps, determine if they are injective or/and surjective.

Exercise 4.18

(See Solution 4.19.) Consider the vectors in \({\bf R}^4\)

\[ \begin{align*} v_1 & = (1, 2, -3, -1) \\ v_2 & = (3, 4, 4, 1) \\ v_3 & = (1, 0, 10, 3). \end{align*} \]

Are \(v_1, v_2, v_3\) linearly independent? What is the dimension of the subspace \(U\) of \({\bf R}^4\) that these vectors span?
Find a basis of \({\bf R}^4\) that contains at least 2 of these three vectors.
Define a map \(f : {\bf R}^4 \to {\bf R}^2\) that satisfies \(f(U) = {\bf R}^2\). Can you define a map \(g : {\bf R}^4 \to {\bf R}^3\) that satisfies \(g(U) = {\bf R}^3\)?

Exercise 4.19

(See Solution 4.20.) Decide whether

\[ A = \left ( \begin{array}{ccc} 1 & 3 & -1 \\ 2 & 1 & 5 \\ 1 & -7 & 13 \end{array} \right ) \]

is invertible. If \(A\) is invertible, determine its inverse.

Exercise 4.20

(See Solution 4.21.) Decide whether \(AB = BA\) holds for the following matrices:

\(A = \left ( \begin{array}{cc} 1 & 2 \\ 2 & 1 \end{array} \right )\), \(B = \left ( \begin{array}{cc} 0 & 1 \\ -1 & 1 \end{array} \right )\)
\(A = \left ( \begin{array}{cc} 3 & 0 \\ 0 & 4 \end{array} \right )\), \(B = \left ( \begin{array}{cc} -1 & 0 \\ 0 & 2 \end{array} \right )\)
\(A = \left ( \begin{array}{cc} 1 & x \\ 0 & 1 \end{array} \right )\), \(B = \left ( \begin{array}{cc} y & 1 \\ 0 & y \end{array} \right )\) for two fixed real numbers \(x\) and \(y\)
\(A = \left ( \begin{array}{cc} 3 & 0 \\ 0 & 3 \end{array} \right )\), \(B = \left ( \begin{array}{cc} b_{11} & b_{21} \\ b_{12} & b_{22} \end{array} \right )\) an arbitrary \(2 \times 2\)-matrix
\(A\) an arbitrary matrix, \(B = AA\) (the product of \(A\) with itself, this is also denoted \(A^2\))

Exercise 4.21

(See Solution 4.22.) Determine in each case whether there is a matrix \(A\) satisfying the following condition. If so, is there a unique such matrix or can there be several matrices satisfying the condition? Describe your findings geometrically.

\(A \left ( \begin{array}{c} 1 \\ 0 \end{array} \right ) = \left ( \begin{array}{c} 3 \\ 4 \end{array} \right )\) and \(A \left ( \begin{array}{c} 0 \\ 1 \end{array} \right ) = \left ( \begin{array}{c} 4 \\ 5 \end{array} \right )\)
\(A \left ( \begin{array}{c} 1 \\ 0 \end{array} \right ) = \left ( \begin{array}{c} 3 \\ 4 \end{array} \right )\) and \(A \left ( \begin{array}{c} 2 \\ 0 \end{array} \right ) = \left ( \begin{array}{c} 6 \\ 8 \end{array} \right )\)
\(A \left ( \begin{array}{c} 1 \\ 0 \end{array} \right ) = \left ( \begin{array}{c} 3 \\ 4 \end{array} \right )\) and \(A \left ( \begin{array}{c} 2 \\ 0 \end{array} \right ) = \left ( \begin{array}{c} 3 \\ 4 \end{array} \right )\)

Exercise 4.22

(See Solution 4.23.) Find two \(2 \times 2\)-matrices \(A, B\) such that

\[ AB = 0 \]

(the zero matrix), but \(A \ne 0\) and \(B \ne 0\). (Hint: start with \(A = B = 0\), and then change very few entries.)

Exercise 4.23

(See Solution 4.24.) A square matrix \(A\) is called symmetric if

\[ A = A^T. \]

Determine \(s, t \in {\bf R}\) such that the matrix \(\left ( \begin{array}{cc} 1 & s \\ -2 & t \end{array} \right )\) is symmetric.
Let \(A\) be any square matrix. Prove that \(A + A^T\) is always symmetric. (Hint: Use Lemma 4.91).

Exercise 4.24

(See Solution 4.25.) The trace of a square matrix \(A = (a_{ij})\) is defined to be the sum of the entries on the main diagonal:

\[ {{\mathrm {tr}}}(A) := a_{11} + a_{22} + \dots + a_{nn}. \]

Prove the following statements (if you get stuck with the notation, assume first that \(A = \left ( \begin{array}{cc} a & b \\ c & d \end{array} \right )\) is a \(2 \times 2\)-matrix, then \({{\mathrm {tr}}} (A) = a + d\)):

\({{\mathrm {tr}}}(A)={{\mathrm {tr}}}(A^T)\),
\({{\mathrm {tr}}}(AB) = {{\mathrm {tr}}}(BA)\) (for another square matrix \(B\) of the same size). This is noteworthy since \(AB \ne BA\) in general!
\({{\mathrm {tr}}} (A+B) = {{\mathrm {tr}}}(A) + {{\mathrm {tr}}} (B)\) (for another square matrix \(B\) of the same size), \({{\mathrm {tr}}} (rA) = r {{\mathrm {tr}}}(A)\) (for \(r \in {\bf R}\)).
(optional, slightly more challenging) Prove there is no matrix \(B\) such that \(AB-BA = {\mathrm {id}}\).

Exercise 4.25

(See Solution 4.26.) Let \(A = \left ( \begin{array}{cc} a & b \\ c & d \end{array} \right )\) be an arbitrary \(2 \times 2\)-matrix and \(r \in {\bf R}\). Recall that the scalar multiple \(rA = \left ( \begin{array}{cc} ra & rb \\ rc & rd \end{array} \right )\). Find a \(2 \times 2\)-matrix \(R\) such that the matrix product \(RA\) equals the scalar multiple:

\[ RA = rA. \]

Exercise 4.26

(See Solution 4.27.)

Let

\[ A = \left ( \begin{array}{ccc} 0 & a & b \\ 0 & 0 & c \\ 0 & 0 & 0 \end{array} \right ) \]

be a so-called strictly upper triangular matrix (of size \(3 \times 3\)). Compute \(A^2\) and prove that \(A^3 = 0\).

Make a (sensible) similar statement for \(n \times n\)-matrices (cf. Proposition 5.20 for the definition of upper triangular matrices in general).

Exercise 4.27

(See Solution 4.28.) Consider the identity map

\[ {\mathrm {id}} : {\bf R}^2 \to {\bf R}^2, (x, y) \mapsto (x,y). \]

Consider the standard basis \(e_1 = (1,0)\) and \(e_2 = (0,1)\) of the domain, and the basis comprised of \(v_1 = (1,-3)\) and \(v_2 = (2,1)\) on the codomain.

Compute the base change matrix of \({\mathrm {id}}\) with respect to these bases.
Use it to compute the coordinates of \((2,-5)\) in terms of the basis \(v_1, v_2\).

Exercise 4.28

(See Solution 4.29.) Consider the identity map

\[ {\mathrm {id}} : {\bf R}^3 \to {\bf R}^3 \]

and the basis \(v_1 = (1,0,-1)\), \(v_2 = (2,1,1)\), \(v_3 = (-1,-1,7)\) on the domain and the standard basis on the codomain. Compute the base change matrix with respect to these bases.

Find the base change matrix from the standard basis \(e_1, e_2, e_3\) in \({\bf R}^3\) to the basis \(v_1 = (1,1,2)\), \(v_2 =(1,1,3)\), \(v_3 = (7,-1,0)\).

Exercise 4.30

(See Solution 4.30.) Consider the linear map

\[ f: {\bf R}^2 \to {\bf R}^3, v \mapsto Av, \]

where \(A = \left ( \begin{array}{cc} 2 & 1 \\ 0 & 1 \\ -3 & 1 \end{array} \right )\). Compute the matrix \(B\) of \(f\) with respect to the basis \(\underline v = \{v_1=(1,-1), v_2=(3,-1)\}\) in \({\bf R}^2\), and the basis \(\underline t = \{t_1=(1,0,1), t_2 = (2,1,1), t_3=(-1,-1,-1)\}\) in \({\bf R}^3\).

Hint: We may consider the following diagram:

\[ {\bf R}^2_{\underline v} \xrightarrow[H]{{\mathrm {id}}} {\bf R}^2_{\underline e} \xrightarrow[A]{f} {\bf R}^3_{\underline e} \xrightarrow[K]{{\mathrm {id}}} {\bf R}^3_{\underline t}. \]

Here the subscripts at \({\bf R}^2\) indicate which basis we consider. The matrices \(H\) and \(K\) are the base change matrices from the basis \(\underline v\) to the standard basis \(\underline e\), resp. from the standard basis \(\underline e\) to the basis \(\underline t\). Then

\[ B = K \cdot A \cdot H. \]

Consider the linear map

\[ f : {\bf R}^3 \to {\bf R}^3 \]

which in the standard basis (on both the domain and the codomain) is given by

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

Compute the matrix of \(f\) with respect to the basis

\[ v_1 = (0,-1,1), v_2 = (0,1,0), v_3 = (-1,0,1). \]

(on the domain and the codomain).

Exercise 4.32

(See Solution 4.31.) Consider the linear map

\[ f: {\bf R}^2 \to {\bf R}^2, \]

which is given by the matrix \(A = \left ( \begin{array}{cc} 6 & -1 \\ 2 & 3 \end{array} \right )\) with respect to the standard basis in the domain and the codomain.

Find its matrix with respect to the basis \(v_1 = (1,1)\), \(v_2 = (1,2)\) both in the domain and the codomain.

Exercise 4.33

(See Solution 4.32.) Let \(A = \left ( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right )\). Determine the vectors \(x = \left ( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right )\) such that

\[ A x = x. \]

Exercise 4.34

(See Solution 4.33.) Find, if possible the vectors \(x = \left ( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right )\) such that

\[ \left ( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 4 & 2 \\ 0 & 2 & 1 \end{array} \right ) x = 5 x. \]

Exercise 4.35

(See Solution 4.34.) Consider the matrix \(A = \left ( \begin{array}{cc} 3 & 0 \\ 8 & -1 \end{array} \right )\) which represents \(f: {\bf R}^2 \to {\bf R}^2\) with respect to the standard basis. Find the matrix of \(f\) with respect to the basis

\[ \underline v = \{v_1 = (2,1), v_2 = (0,1) \}. \]

For \(A\) and \(f\) as in Exercise 4.35, consider now the basis

\[ \underline v = \{v_1 = (1,2), v_2 = (0,1) \}. \]

Compute the matrix of \(f\) with respect to that basis.

Exercise 4.37

(See Solution 4.35.) Let \(f: {\bf R}^3 \to {\bf R}^3\) be the map whose matrix with respect to the standard basis is

\[ \left ( \begin{array}{ccc} -1 & 1 & 0 \\ 0 & 2 & 0 \\ 1 & -1 & -2 \end{array} \right ). \]

Compute the matrix with respect to the basis

\[ \underline v = \{v_1 = (0,0,1), v_2 = (2,6,-1), v_3 = (1,0,1) \}. \]

Exercise 4.38

(See Solution 4.36.) Let \(f: {\bf R}^3 \to {\bf R}^3\) be the linear map whose matrix with respect to the standard basis is

\[ A = \left ( \begin{array}{ccc} 1 & -1 & 0 \\ 1 & 2 & 1 \\ 2 & 1 & 3 \end{array} \right ). \]

Find a basis of the solution space \(L\) of the linear system

\[ A \left ( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right ) = 3 \left ( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right ). \]

(As a forecast to terminology introduced later, this solution space is the so-called eigenspace of \(A\) for the eigenvalue 3, cf. Definition and Lemma 6.11.)

Complete the basis of \(L\) (which is a subspace of \({\bf R}^3\)) to a basis of \({\bf R}^3\), and compute the matrix of \(f\) with respect to this basis.

Exercise 4.39

(See Solution 4.37.) Consider the linear map \(f: {\bf R}^3 \to {\bf R}^3\) whose kernel is \(L((1,0,1))\) and such that

\[ f(0,3,-1) = (0,3,-1), f(0,0,1) = (0,0,2). \]

Compute its matrix with respect to the standard basis.

Consider the linear map \(f: {\bf R}^3 \to {\bf R}^3\) such that

\[ f(1,1,1)=3\cdot (1,1,1), f(2,0,1) = (-4,0,-2), f(0,1,3) = (0,2,6). \]

Show that the vectors

\[ \underline v = \{(1,1,1), (2,0,1), (0,1,3)\} \]

form a basis of \({\bf R}^3\). (Note that for each of these three vectors, one has \(f(v_i) = \lambda_i v\), with \(\lambda_1 = 3\) etc. Therefore, the basis is an example of a so-called eigenbasis, cf. Definition 6.17.)

Compute the matrix of \(f\) with respect to that basis.
Compute the matrix of \(f\) with respect to the standard basis.

The following two exercises are all concerned with linear systems of the form

\[ A x = \lambda x, \]

where \(A\) is a certain square matrix, \(x\) is a vector and \(\lambda \in {\bf R}\) a real number. We will study these systems systematically in §Chapter 6.

Exercise 4.41

(See Solution 4.38.) Find the solutions of the linear system

\[ \left ( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 4 & 2 \\ 0 & 2 & 1 \end{array} \right ) \left ( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right ) = 5 \left ( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right ). \]

Solve the system

\[ \left ( \begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right ) \left ( \begin{array}{c} a \\ b \\ c \\ d \end{array} \right ) = 1 \cdot \left ( \begin{array}{c} a \\ b \\ c \\ d \end{array} \right ). \]

Exercise 4.43

(See Solution 4.39.) Consider the vectors \(v_1 = (1,0,-1)\), \(v_2 = (1,1,0)\), \(v_3 = (1,0,-2) \in {\bf R}^3\).

Let \(f\colon {\bf R}^3 \to {\bf R}^3\) be the linear map such that \(f(v_1) = (3,0,-5)\), \(f(v_2) = (2,2,0)\) (in the terminology of Definition 6.1, \(v_2\)is an eigenvector with eigenvalue \(2\)) and \(f(v_3) = (5,2,-5)\).

Determine the matrix \(A\) of \(f\) with respect to the basis \(v_1, v_2, v_3\) (both in the domain and the codomain).
Determine the matrix \(B\) of \(f\) with respect to the standard basis of \({\bf R}^3\).
Compute a basis of the kernel and of the image of \(f\).
Determine for which value of \(t\) is the vector \(w = (3,t,-5)\) in the image of \(f\). For such \(t\), compute the preimage \(f^{-1}(w)\).

Exercise 4.44

(See Solution 4.40.) Let \(f\colon {\bf R}^3 \to {\bf R}^4\) be the following linear map:

\[ f(x,y,z) = (x-y+2z,\; -2x+3y-z,\; y+3z,\; -x+3y+tz) \]

Determine the dimension of the image of \(f\) (depending on the parameter \(t \in {\bf R}\)).
Are there values of \(t\) for which \(f\) is surjective? If so, what are these values of \(t\)? Are there values of \(t\) for which \(f\) is injective? If so, what are they?
For the value of \(t\) for which the rank of \(f\) is 2, compute a basis of \(\ker f\) and of \(\Im f\).
Are there values of \(t\) for which the vector \(w = (1,1,0,1)\) belongs to the image of \(f\)?
We now put \(t = 0\). Is there a linear map \(g\colon {\bf R}^4 \to {\bf R}^3\) such that the composite \(g\circ f\colon {\bf R}^3 \to {\bf R}^3\) is the identity?

Exercise 4.45

(See Solution 4.41.) Let \(f\colon {\bf R}^3 \to {\bf R}^4\) be the following linear map:

\[ f(x,y,z) = (-y+2z,\; 2x+y,\; -x+2y-5z,\; x+2y-3z) \]

Compute a basis of \(\ker f\) and a basis of \({\operatorname{im}\ } f\).
Let \(W \subset {\bf R}^4\) be the subspace defined by the equation \(x_2 - 3x_3 = 0\). Compute the dimension and a basis of \(W\).
We put \(U = {\operatorname{im}\ } f\). Compute a basis of \(U \cap W\) and a basis of \(U+W\).
Determine for which values of \(a\) and \(b\) there exists a vector \(v \in {\bf R}^3\) such that \(f(v) = (a,4,3,b)\).

Exercise 4.46

(See Solution 4.42.) Let \(U\) be the subspace of \({\bf R}^4\) generated by the vectors \(u_1 = (1,-2,0,1)\), \(u_2 = (-1,1,2,0)\), \(u_3 = (0,3,1,-1)\). Let \(W\subset {\bf R}^4\) be the subspace with equations

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

Compute the dimension and a basis of \(W\).
Compute a basis of \(U \cap W\) and a basis of \(U + W\).
Compute a basis of a subspace \(U' \subset U\) such that \(U' \oplus W = {\bf R}^4\).

Exercise 4.47

(See Solution 4.43.) Consider the matrix

\[ A = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 4a^2 & -1 \\ -1 & -2 & a \end{pmatrix} \]

Determine the rank of \(A\) for all values of the parameter \(a \in {\bf R}\).
Set \(a = 1\) and consider the column vector \(B = (b_1,b_2,b_3)\). What relation must hold among the variables \(b_1,b_2,b_3\) so that the system \(AX = B\) has solutions?
Set \(a = -1\) and consider the column vector \(C = (2,7,c)\). Find the value of \(c\) for which the system \(AX = C\) has solutions and, for that value of \(c\), find all solutions of the system.