Systems of Linear Equations
---------------------------
**Exercise 0.** :math:`\\`
Write down the following problems in a form of a system of linear equations and find its solutions (by any means).
#. | Andrew is two times older than Peter, and the sum of their age equals 33.
| How old are Andrew and Peter?
#. | The points :math:`\ P_1=(2,5)\ ` and :math:`\ P_2=(3,7)\ ` lie on a line given by the equation
:math:`\ y=ax+b.\ `
| Find the parameters :math:`\ a\ ` and :math:`\ b.`
#. | The parabola passes through the points :math:`\ P_1=(1,4),\ P_2=(2,8)\ ` and :math:`\ P_3=(3,14).\ `
| Find the coefficients :math:`\ a,\,b,\,c\ ` which determine the equation :math:`\ y=ax^2+bx+c\ ` of this parabola.
**Exercise 1.** :math:`\\`
Perform elementary operations on the equations (cf. :ref:`Gaussian Elimination`)
or rows of the corresponding matrices (cf. :ref:`Practical Elimination in Sage`), without applying functions ``solve()``,
:math:`\,` ``X.solve_right()``, :math:`\,` ``X\y``, :math:`\,` ``X.rref()``, :math:`\,`
in order to solve the following systems of equations over the field :math:`\ Q:`
#. .. math::
:nowrap:
\begin{alignat*}{3}
x_1 & \ -\ & x_2 & \ =\ & 0 \\
-\,x_1 & \ +\ & 2\,x_2 & \ =\ & \ \textstyle\frac{1}{36}
\end{alignat*}
.. (1/36, 1/36), rank A: 2
#. .. math::
:nowrap:
\begin{alignat*}{3}
-\,2\,x_1 & \ -\ & 3\,x_2 & \ =\ & 0 \\
3\,x_1 & \ +\ & 4\,x_2 & \ =\ & \ \textstyle\frac{16}{7}
\end{alignat*}
.. (48/7, -32/7), rank A: 2
#. .. math::
:nowrap:
\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*}
.. (16, 71/5, 83/10), rank A: 3
#. .. math::
:nowrap:
\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*}
.. (2, -13, -4), rank A: 3
#. .. math::
:nowrap:
\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*}
.. (223/21, 157/21, 11/7), rank A: 3
#. .. math::
:nowrap:
\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*}
.. (6, 20, -27/2, -7/2), rank A: 4
**Exercise 2.** :math:`\,`
Now use all necessary functions of Sage (c.f. :ref:`Example with Discussion`)
to solve real linear problems of the form :math:`\ \boldsymbol{A}\,\boldsymbol{x}=\boldsymbol{b}\ \,`
for the following data:
1. .. math::
\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]\,;
2. .. math::
\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]\,;
.. .. math::
\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}
0 \\ 0 \\ 0 \\ 0
\end{array}\right]\,;
3. .. math::
\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]\,.
A basis of the solution space of the homogeneous linear problem
:math:`\ \boldsymbol{A}\,\boldsymbol{x}=\boldsymbol{0}\ ` is called a :math:`\,`
*fundamental set of solutions* :math:`\,` for this problem.
**Exercise 3.** :math:`\,`
Find a fundamental set of solutions of the homogeneous linear problem
over :math:`\ Q\ ` with coefficient matrix
.. math::
\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]\,.
**Exercise 4.** :math:`\,`
Find a homogeneous system of equations consisting of
:math:`\,` a.) two :math:`\,` b.) three :math:`\,` equations so that the vectors
.. math::
\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]
comprise its fundamental set of solutions.
.. (4.4.30)
**Exercise 5.** :math:`\,`
Does there exist a homogeneous system of linear equations whose fundamental set of solutions
is given both by the vectors
:math:`\ (\boldsymbol{x}_1,\boldsymbol{x}_2,\boldsymbol{x}_3)\ ` and
:math:`\ (\boldsymbol{y}_1,\boldsymbol{y}_2,\boldsymbol{y}_3),\ ` where
.. math::
\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}
.. (4.4.31)
**Exercise 6.** :math:`\\`
Does there exist :math:`\ \lambda\in Q\ ` for which the followng system of linear equations
.. math::
:nowrap:
\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 :math:`\ Q\ `?
**Hint.** The (negative) answer may be given by computation of only one
determinant of rank 3.
**Exercise 7.**
For which :math:`\ \lambda\in R\ ` the following system of linear equations over the field :math:`\ R\ `
has a solution? :math:`\,` Find this solution.
.. math::
:nowrap:
\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 :math:`\ \lambda`:
#. .. math::
:nowrap:
\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*}
#. .. math::
:nowrap:
\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*}
#. .. math::
:nowrap:
\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.** :math:`\,`
Determine all the values of :math:`\ \lambda\in R\ ` for which
a vector :math:`\ \boldsymbol{b}\ ` may be expressed as a linear combination of vectors
:math:`\ \boldsymbol{a}_1,\,\boldsymbol{a}_2,\,\boldsymbol{a}_3:`
.. math::
\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}
**Hint.** :math:`\,`
Study existence of solutions for a linear problem given in a column form
.. math::
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 :math:`\ x\ `
of degree (at most) :math:`\ n.\ `
What is the dimension of the subspace consisting of polynomials satisfying the conditions
:math:`\ w(x_1)=w(x_2)=\ldots=w(x_k)=0,\ ` where :math:`\ x_1,x_2,\ldots,x_k\ `
are pairwise distinct numbers :math:`\ (k\le n).`
.. (4.4.28) Odpowiedź: n+1-k.
**Hnt.**
Use a formula for the Vandermonde determinant (see equation :eq:`Vandermonde`).
**Exercise 11.**
Find a basis for a vector space of real polynomials of one variable of degree (at most) 5
satisfying the conditions :math:`\ w(0)=w(1)=w(2)=w(3)=0.`
.. (4.4.29)`