Systems of Linear Equations

Exercise 0. \(\\\) Write down the following problems in a form of a system of linear equations and find its solutions (by any means).

1. Andrew is two times older than Peter, and the sum of their age equals 33.

   How old are Andrew and Peter?
2. The points \(\ P_1=(2,5)\ \) and \(\ P_2=(3,7)\ \) lie on a line given by the equation \(\ y=ax+b.\ \)

   Find the parameters \(\ a\ \) and \(\ b.\)
3. The parabola passes through the points \(\ P_1=(1,4),\ P_2=(2,8)\ \) and \(\ P_3=(3,14).\ \)

   Find the coefficients \(\ a,\,b,\,c\ \) which determine the equation \(\ y=ax^2+bx+c\ \) of this parabola.

Exercise 1. \(\\\) Perform elementary operations on the equations (cf. Gaussian Elimination) or rows of the corresponding matrices (cf. Practical Elimination in Sage), without applying functions solve(), \(\,\) X.solve_right(), \(\,\) X\y, \(\,\) X.rref(), \(\,\) in order to solve the following systems of equations over the field \(\ Q:\)

1. \begin{alignat*}{3} x_1 & \ -\ & x_2 & \ =\ & 0 \\ -\,x_1 & \ +\ & 2\,x_2 & \ =\ & \ \textstyle\frac{1}{36} \end{alignat*}
2. \begin{alignat*}{3} -\,2\,x_1 & \ -\ & 3\,x_2 & \ =\ & 0 \\ 3\,x_1 & \ +\ & 4\,x_2 & \ =\ & \ \textstyle\frac{16}{7} \end{alignat*}
3. \begin{alignat*}{4} 16\,x_1 & \ +\ & 41\,x_2 & \ -\ & 101\,x_3 & \ =\ & -\,\textstyle\frac{1}{10} \\ x_1 & \ +\ & 3\,x_2 & \ -\ & 7\,x_3 & \ =\ & \ \textstyle\frac{1}{2} \\ -\,5\,x_1 & \ -\ & 13\,x_2 & \ +\ & 32\,x_3 & \ =\ & 1 \end{alignat*}
4. \begin{alignat*}{4} -\,x_1 & \ -\ & 3\,x_2 & \ +\ & 9\,x_3 & \ =\ & 1 \\ x_1 & \ \ & & \ +\ & x_3 & \ =\ & -\,2 \\ -\,2\,x_1 & \ -\ & 2\,x_2 & \ +\ & 5\,x_3 & \ =\ & 2 \end{alignat*}
5. \begin{alignat*}{4} -\,11\,x_1 & \ +\ & 44\,x_2 & \ -\ & 135\,x_3 & \ =\ & 0 \\ 2\,x_1 & \ -\ & 7\,x_2 & \ +\ & 20\,x_3 & \ =\ & \textstyle\frac{1}{3} \\ 4\,x_1 & \ -\ & 16\,x_2 & \ +\ & 49\,x_3 & \ =\ & -\ \textstyle\frac{1}{7} \end{alignat*}
6. \begin{alignat*}{5} x_1 &\ +\ & 3\,x_2 &\ -\ & x_3 &\ +\ & 23 \,x_4 &\ =\ & -\,1 \\ 4\,x_1 &\ +\ & 13\,x_2 &\ -\ & 3\,x_3 &\ +\ & 93 \,x_4 &\ =\ & -\,1 \\ -\,5\,x_1 &\ -\ & 17\,x_2 &\ +\ & 4\,x_3 &\ -\ & 121 \,x_4 &\ =\ & -\ \textstyle\frac{1}{2} \\ &\ \ & x_2 &\ +\ & 3\,x_3 &\ -\ & 6 \,x_4 &\ =\ & \textstyle\frac{1}{2} \end{alignat*}

Exercise 2. \(\,\) Now use all necessary functions of Sage (c.f. Example with Discussion) to solve real linear problems of the form \(\ \boldsymbol{A}\,\boldsymbol{x}=\boldsymbol{b}\ \,\) for the following data:

1. \[\begin{split}\boldsymbol{A}\ =\ \left[\begin{array}{rrrr} 1 & -5 & 9 & 11 \\ 1 & -4 & 8 & 9 \\ -3 & 15 & -26 & -33 \\ -2 & 7 & -10 & -16 \end{array}\right]\,,\qquad \boldsymbol{b}\ =\ \left[\begin{array}{r} -1 \\ 0 \\ 4 \\ 4 \end{array}\right]\,;\end{split}\]
2. \[\begin{split}\boldsymbol{A}\ =\ \left[\begin{array}{rrrr} 1 & 4 & 5 & -1 \\ -3 & -12 & -14 & 3 \\ 3 & 12 & 19 & -3 \\ -2 & -8 & -12 & 2 \end{array}\right]\,,\qquad \boldsymbol{b}\ =\ \left[\begin{array}{r} 13 \\ -38 \\ 43 \\ -28 \end{array}\right]\,;\end{split}\]

3. \[\begin{split}\boldsymbol{A}\ =\ \left[\begin{array}{rrr} 1 & -5 & -11 \\ 2 & -9 & -20 \\ 4 & -16 & -36 \end{array}\right]\,,\qquad \boldsymbol{b}\ =\ \left[\begin{array}{r} -1 \\ -5 \\ 1 \end{array}\right]\,.\end{split}\]

A basis of the solution space of the homogeneous linear problem \(\ \boldsymbol{A}\,\boldsymbol{x}=\boldsymbol{0}\ \) is called a \(\,\) fundamental set of solutions \(\,\) for this problem.

Exercise 3. \(\,\) Find a fundamental set of solutions of the homogeneous linear problem over \(\ Q\ \) with coefficient matrix

\[\begin{split}\boldsymbol{A}\ =\ \left[\begin{array}{rrrr} 1 & 4 & 5 & -1 \\ -3 & -12 & -14 & 3 \\ 3 & 12 & 19 & -3 \\ -2 & -8 & -12 & 2 \end{array}\right]\,.\end{split}\]

Exercise 4. \(\,\) Find a homogeneous system of equations consisting of \(\,\) a.) two \(\,\) b.) three \(\,\) equations so that the vectors

\[\begin{split}\left[\begin{array}{r} 1 \\ 4 \\ -2 \\ 2 \\ -1 \end{array}\right]\,,\quad \left[\begin{array}{r} 3 \\ 13 \\ -1 \\ 2 \\ 1 \end{array}\right]\,,\quad \left[\begin{array}{r} 2 \\ 7 \\ -8 \\ 4 \\ -5 \end{array}\right]\end{split}\]

comprise its fundamental set of solutions.

Exercise 5. \(\,\) Does there exist a homogeneous system of linear equations whose fundamental set of solutions is given both by the vectors \(\ (\boldsymbol{x}_1,\boldsymbol{x}_2,\boldsymbol{x}_3)\ \) and \(\ (\boldsymbol{y}_1,\boldsymbol{y}_2,\boldsymbol{y}_3),\ \) where

\[\begin{split}\begin{array}{lll} \boldsymbol{x}_1= \left[\begin{array}{r} 2 \\ 3 \\ 1 \\ 2 \end{array}\right], & \boldsymbol{x}_2= \left[\begin{array}{r} 1 \\ 1 \\ -2 \\ -2 \end{array}\right], & \boldsymbol{x}_3= \left[\begin{array}{r} 3 \\ 4 \\ 2 \\ 1 \end{array}\right], \\ \\ \boldsymbol{y}_1= \left[\begin{array}{r} 1 \\ 0 \\ 2 \\ -5 \end{array}\right], & \boldsymbol{y}_2= \left[\begin{array}{r} 0 \\ 1 \\ 8 \\ 7 \end{array}\right], & \boldsymbol{y}_3= \left[\begin{array}{r} 4 \\ 5 \\ -2 \\ 0 \end{array}\right]. \end{array}\end{split}\]

Exercise 6. \(\\\) Does there exist \(\ \lambda\in Q\ \) for which the followng system of linear equations

\begin{alignat*}{4} x_1 & \ +\ & 2\,x_2 & \ +\ & 3\,\lambda\,x_3 & \ =\ & -1 \\ x_1 & \ +\ & x_2 & \ -\ & x_3 & \ =\ & 1 \\ \,2\,x_1 & \ +\ & x_2 & \ +\ & 5\,x_3 & \ =\ & 3 \end{alignat*}

has infinitely many solutions over \(\ Q\ \)?

Hint. The (negative) answer may be given by computation of only one determinant of rank 3.

Exercise 7. For which \(\ \lambda\in R\ \) the following system of linear equations over the field \(\ R\ \) has a solution? \(\,\) Find this solution.

\begin{alignat*}{5} 2\,x_1 &\ -\ & x_2 &\ +\ & x_3 &\ +\ & x_4 &\ =\ & 1 \\ x_1 &\ +\ & 2\,x_2 &\ -\ & x_3 &\ +\ & 4\,x_4 &\ =\ & 2 \\ x_1 &\ +\ & 7\,x_2 &\ -\ & 4\,x_3 &\ +\ & 11\,x_4 &\ =\ & \lambda \end{alignat*}

Exercise 8. Describe a solution space of the following systems of equations depending on \(\ \lambda\):

1. \begin{alignat*}{4} 3\,x_1 & \ +\ & 2\,x_2 & \ +\ & x_3 & \ =\ & -1 \\ 7\,x_1 & \ +\ & 6\,x_2 & \ +\ & 5\,x_3 & \ =\ & \lambda \\ 5\,x_1 & \ +\ & 4\,x_2 & \ +\ & 3\,x_3 & \ =\ & 2 \end{alignat*}
2. \begin{alignat*}{4} \lambda\,x_1 & \ +\ & x_2 & \ +\ & x_3 & \ =\ & 0 \\ 5\,x_1 & \ +\ & x_2 & \ -\ & 2\,x_3 & \ =\ & 2 \\ -2\,x_1 & \ -\ & 2\,x_2 & \ +\ & x_3 & \ =\ & -3 \end{alignat*}
3. \begin{alignat*}{4} x_1 & \ +\ & x_2 & \ +\ & \lambda\,x_3 & \ =\ & 1 \\ x_1 & \ +\ & \lambda\,x_2 & \ +\ & x_3 & \ =\ & 1 \\ \lambda\,x_1 & \ +\ & x_2 & \ +\ & x_3 & \ =\ & 1 \end{alignat*}

Exercise 9. \(\,\) Determine all the values of \(\ \lambda\in R\ \) for which a vector \(\ \boldsymbol{b}\ \) may be expressed as a linear combination of vectors \(\ \boldsymbol{a}_1,\,\boldsymbol{a}_2,\,\boldsymbol{a}_3:\)

\[\begin{split}\begin{array}{lllll} 1.) & \qquad \boldsymbol{a}_1=\left[\begin{array}{r} 2 \\ 3 \\ 5 \end{array}\right], & \boldsymbol{a}_2=\left[\begin{array}{r} 3 \\ 7 \\ 8 \end{array}\right], & \boldsymbol{a}_3=\left[\begin{array}{r} 1 \\ -6 \\ 1 \end{array}\right], & \quad \boldsymbol{b} = \left[\begin{array}{r} 7 \\ -2 \\ \lambda \end{array}\right]; \\ \\ 2.) & \qquad \boldsymbol{a}_1=\left[\begin{array}{r} 4 \\ 4 \\ 3 \end{array}\right], & \boldsymbol{a}_2=\left[\begin{array}{r} 7 \\ 2 \\ 1 \end{array}\right], & \boldsymbol{a}_3=\left[\begin{array}{r} 4 \\ 1 \\ 6 \end{array}\right], & \quad \boldsymbol{b} = \left[\begin{array}{r} 5 \\ 9 \\ \lambda \end{array}\right]; \\ \\ 3.) & \qquad \boldsymbol{a}_1=\left[\begin{array}{r} 3 \\ 2 \\ 6 \end{array}\right], & \boldsymbol{a}_2=\left[\begin{array}{r} 7 \\ 3 \\ 9 \end{array}\right], & \boldsymbol{a}_3=\left[\begin{array}{r} 5 \\ 1 \\ 3 \end{array}\right], & \quad \boldsymbol{b} = \left[\begin{array}{r} \lambda \\ 2 \\ 5 \end{array}\right]; \\ \\ 4.) & \qquad \boldsymbol{a}_1=\left[\begin{array}{r} 3 \\ 2 \\ 5 \end{array}\right], & \boldsymbol{a}_2=\left[\begin{array}{r} 2 \\ 4 \\ 7 \end{array}\right], & \boldsymbol{a}_3=\left[\begin{array}{r} 5 \\ 6 \\ \lambda \end{array}\right], & \quad \boldsymbol{b} = \left[\begin{array}{r} 1 \\ 3 \\ 5 \end{array}\right]. \end{array}\end{split}\]

Hint. \(\,\) Study existence of solutions for a linear problem given in a column form

\[x_1\,\boldsymbol{a}_1+x_2\,\boldsymbol{a}_2+x_3\,\boldsymbol{a}_3=\boldsymbol{b}.\]

Exercise 10. Consider a vector space of real polynomials of one variable \(\ x\ \) of degree (at most) \(\ n.\ \) What is the dimension of the subspace consisting of polynomials satisfying the conditions \(\ w(x_1)=w(x_2)=\ldots=w(x_k)=0,\ \) where \(\ x_1,x_2,\ldots,x_k\ \) are pairwise distinct numbers \(\ (k\le n).\)

Hnt. Use a formula for the Vandermonde determinant (see equation (2)).

Exercise 11. Find a basis for a vector space of real polynomials of one variable of degree (at most) 5 satisfying the conditions \(\ w(0)=w(1)=w(2)=w(3)=0.\)