Ćwiczenia I (15 X 2020). Układy równań

1. Rozwiąż układ równań

$$\left\{\begin{array}{rrr} 2 x &+& 3 y &=& 1 \\ 3 x &+& y &=& 0 \\ \end{array}\right.$$
In [1]:
from gal import IMatrix
This module, meant for **educational purposes only**, supports learning of basics of linear algebra.

It was created to supplement a "Linear algebra and geometry I" course taught during winter semester 2020
at the University of Warsaw Mathematics Department.

AUTHORS:
    Andrzej Nagórko, Jarosław Wiśniewski

In [3]:
A = IMatrix([[2, 3, 1], [3,1,0]], separate=1, var=['x', 'y'])
show(A.as_equations())
In [4]:
A.as_equations().rescale_row(0, -3)
Out[4]:
\[\left\{\begin{array}{ccccl} 2 x &+& 3 y &=& 1 \\ 3 x &+& y &=& 0 \\ \end{array}\right.\begin{array}{c} /\cdot -3\\\ \end{array}\rightarrow\left\{\begin{array}{ccccl} -6 x &-& 9 y &=& -3 \\ 3 x &+& y &=& 0 \\ \end{array}\right.\]
In [5]:
A.as_equations().rescale_row(1, 2)
Out[5]:
\[\left\{\begin{array}{ccccl} -6 x &-& 9 y &=& -3 \\ 3 x &+& y &=& 0 \\ \end{array}\right.\begin{array}{c} \ \\/\cdot 2 \end{array}\rightarrow\left\{\begin{array}{ccccl} -6 x &-& 9 y &=& -3 \\ 6 x &+& 2 y &=& 0 \\ \end{array}\right.\]
In [6]:
A.as_equations().add_multiple_of_row(0, 1, 1)
Out[6]:
\[\left\{\begin{array}{ccccl} -6 x &-& 9 y &=& -3 \\ 6 x &+& 2 y &=& 0 \\ \end{array}\right.\begin{array}{c} /+1 \cdot w_2\\\ \end{array}\rightarrow\left\{\begin{array}{ccccl} && -7 y &=& -3 \\ 6 x &+& 2 y &=& 0 \\ \end{array}\right.\]
In [7]:
A.as_equations().rescale_row(0, -1/7)
Out[7]:
\[\left\{\begin{array}{ccccl} && -7 y &=& -3 \\ 6 x &+& 2 y &=& 0 \\ \end{array}\right.\begin{array}{c} /\cdot -\frac{1}{7}\\\ \end{array}\rightarrow\left\{\begin{array}{ccccl} && y &=& \frac{3}{7} \\ 6 x &+& 2 y &=& 0 \\ \end{array}\right.\]
In [8]:
A.as_equations().add_multiple_of_row(1, 0, -2)
Out[8]:
\[\left\{\begin{array}{ccccl} && y &=& \frac{3}{7} \\ 6 x &+& 2 y &=& 0 \\ \end{array}\right.\begin{array}{c} \ \\/-2 \cdot w_1 \end{array}\rightarrow\left\{\begin{array}{ccccl} && y &=& \frac{3}{7} \\ 6 x && &=& -\frac{6}{7} \\ \end{array}\right.\]
In [9]:
A.as_equations().rescale_row(1, 1/6)
Out[9]:
\[\left\{\begin{array}{ccccl} && y &=& \frac{3}{7} \\ 6 x && &=& -\frac{6}{7} \\ \end{array}\right.\begin{array}{c} \ \\/\cdot \frac{1}{6} \end{array}\rightarrow\left\{\begin{array}{ccccl} && y &=& \frac{3}{7} \\ x&& &=& -\frac{1}{7} \\ \end{array}\right.\]
In [10]:
A.as_equations().swap_rows(0, 1)
Out[10]:
\[\left\{\begin{array}{ccccl} && y &=& \frac{3}{7} \\ x&& &=& -\frac{1}{7} \\ \end{array}\right.\begin{array}{c} /\leftarrow w_2\\/\leftarrow w_1 \end{array}\rightarrow\left\{\begin{array}{ccccl} x&& &=& -\frac{1}{7} \\ && y &=& \frac{3}{7} \\ \end{array}\right.\]

2. Rozwiąż układ równań $$\left\{\begin{array}{cccccl} x &+& y && &=& 1 \\ x &+& 2 y &-& 3 z &=& -3 \\ 2 x &+& 4 y &+& z &=& 1 \\ \end{array}\right.$$

In [17]:
# (x, y, z) = (2, -1, 1)

A = IMatrix([[1,1,0,1], [1,2,-3,-3], [2,4,1,1]], separate=1, var=['x','y','z'])
show(A.as_equations())
In [18]:
A.as_equations().add_multiple_of_row(2, 1, -2)
Out[18]:
\[\left\{\begin{array}{ccccccl} x&+& y&& &=& 1 \\ x &+& 2 y &-& 3 z &=& -3 \\ 2 x &+& 4 y &+& z &=& 1 \\ \end{array}\right.\begin{array}{c} \ \\\ \\/-2 \cdot w_2 \end{array}\rightarrow\left\{\begin{array}{ccccccl} x&+& y&& &=& 1 \\ x &+& 2 y &-& 3 z &=& -3 \\ && && 7 z &=& 7 \\ \end{array}\right.\]
In [19]:
A.as_equations().rescale_row(2, 1/7)
Out[19]:
\[\left\{\begin{array}{ccccccl} x&+& y&& &=& 1 \\ x &+& 2 y &-& 3 z &=& -3 \\ && && 7 z &=& 7 \\ \end{array}\right.\begin{array}{c} \ \\\ \\/\cdot \frac{1}{7} \end{array}\rightarrow\left\{\begin{array}{ccccccl} x&+& y&& &=& 1 \\ x &+& 2 y &-& 3 z &=& -3 \\ && && z &=& 1 \\ \end{array}\right.\]
In [21]:
show(A.as_equations())
show(A)
In [22]:
A.add_multiple_of_row(1,2,3)
Out[22]:
\[\left[\begin{array}{rrr|r} 1 & 1 & 0 & 1\\ 1 & 2 & -3 & -3\\ 0 & 0 & 1 & 1\\ \end{array}\right]\begin{array}{c} \ \\+3 \cdot w_3\\\ \end{array}\rightarrow\left[\begin{array}{rrr|r} 1 & 1 & 0 & 1\\ 1 & 2 & 0 & 0\\ 0 & 0 & 1 & 1\\ \end{array}\right]\]
In [24]:
show(A.as_equations())
In [25]:
A.add_multiple_of_row(1, 0, -1)
Out[25]:
\[\left[\begin{array}{rrr|r} 1 & 1 & 0 & 1\\ 1 & 2 & 0 & 0\\ 0 & 0 & 1 & 1\\ \end{array}\right]\begin{array}{c} \ \\-1 \cdot w_1\\\ \end{array}\rightarrow\left[\begin{array}{rrr|r} 1 & 1 & 0 & 1\\ 0 & 1 & 0 & -1\\ 0 & 0 & 1 & 1\\ \end{array}\right]\]
In [26]:
A.add_multiple_of_row(0, 1, -1)
Out[26]:
\[\left[\begin{array}{rrr|r} 1 & 1 & 0 & 1\\ 0 & 1 & 0 & -1\\ 0 & 0 & 1 & 1\\ \end{array}\right]\begin{array}{c} -1 \cdot w_2\\\ \\\ \end{array}\rightarrow\left[\begin{array}{rrr|r} 1 & 0 & 0 & 2\\ 0 & 1 & 0 & -1\\ 0 & 0 & 1 & 1\\ \end{array}\right]\]
In [28]:
show(A.as_equations())

3. Rozwiąż układ równań $$\left\{\begin{array}{ccccccl} 3 x &+& y &+& z &=& -1 \\ x&& &+& 2 z &=& -6 \\ && 3 y &+& 2 z &=& 0 \\ \end{array}\right.$$

In [35]:
A = IMatrix([[3,1,1,-1], [1,0,2,-6], [0,3,2,0]], separate=1, var=['x','y','z'])
show(A)
In [36]:
A.add_multiple_of_row(0, 1, -3)
Out[36]:
\[\left[\begin{array}{rrr|r} 3 & 1 & 1 & -1\\ 1 & 0 & 2 & -6\\ 0 & 3 & 2 & 0\\ \end{array}\right]\begin{array}{c} -3 \cdot w_2\\\ \\\ \end{array}\rightarrow\left[\begin{array}{rrr|r} 0 & 1 & -5 & 17\\ 1 & 0 & 2 & -6\\ 0 & 3 & 2 & 0\\ \end{array}\right]\]
In [37]:
A.swap_rows(0,1)
Out[37]:
\[\left[\begin{array}{rrr|r} 0 & 1 & -5 & 17\\ 1 & 0 & 2 & -6\\ 0 & 3 & 2 & 0\\ \end{array}\right]\begin{array}{c} \leftarrow w_2\\\leftarrow w_1\\\ \end{array}\rightarrow\left[\begin{array}{rrr|r} 1 & 0 & 2 & -6\\ 0 & 1 & -5 & 17\\ 0 & 3 & 2 & 0\\ \end{array}\right]\]
In [38]:
A.add_multiple_of_row(2, 1, -3)
Out[38]:
\[\left[\begin{array}{rrr|r} 1 & 0 & 2 & -6\\ 0 & 1 & -5 & 17\\ 0 & 3 & 2 & 0\\ \end{array}\right]\begin{array}{c} \ \\\ \\-3 \cdot w_2 \end{array}\rightarrow\left[\begin{array}{rrr|r} 1 & 0 & 2 & -6\\ 0 & 1 & -5 & 17\\ 0 & 0 & 17 & -51\\ \end{array}\right]\]
In [40]:
A.rescale_row(2, 1/17)
Out[40]:
\[\left[\begin{array}{rrr|r} 1 & 0 & 2 & -6\\ 0 & 1 & -5 & 17\\ 0 & 0 & 17 & -51\\ \end{array}\right]\begin{array}{c} \ \\\ \\\cdot \frac{1}{17} \end{array}\rightarrow\left[\begin{array}{rrr|r} 1 & 0 & 2 & -6\\ 0 & 1 & -5 & 17\\ 0 & 0 & 1 & -3\\ \end{array}\right]\]
In [41]:
show(A.as_equations())
In [42]:
A.add_multiple_of_row(0, 2, -2)
Out[42]:
\[\left[\begin{array}{rrr|r} 1 & 0 & 2 & -6\\ 0 & 1 & -5 & 17\\ 0 & 0 & 1 & -3\\ \end{array}\right]\begin{array}{c} -2 \cdot w_3\\\ \\\ \end{array}\rightarrow\left[\begin{array}{rrr|r} 1 & 0 & 0 & 0\\ 0 & 1 & -5 & 17\\ 0 & 0 & 1 & -3\\ \end{array}\right]\]
In [43]:
A.add_multiple_of_row(1, 2, 5)
Out[43]:
\[\left[\begin{array}{rrr|r} 1 & 0 & 0 & 0\\ 0 & 1 & -5 & 17\\ 0 & 0 & 1 & -3\\ \end{array}\right]\begin{array}{c} \ \\+5 \cdot w_3\\\ \end{array}\rightarrow\left[\begin{array}{rrr|r} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 2\\ 0 & 0 & 1 & -3\\ \end{array}\right]\]
In [44]:
show(A.as_equations())

4. Rozwiąż układ równań $$\left\{\begin{array}{ccccccl} 2 x &+& 3 y &+& 2 z &=& 1 \\ 3 x &+& 4 y &+& 2 z &=& 2 \\ 4 x &+& 2 y &+& 3 z &=& 3 \\ \end{array}\right.$$

In [ ]:
# x = 8/7, y = -1/7, z = -3/7

5. Rozwiąż układ równań $$\left\{\begin{array}{ccccccccl} x&+& y&+& z&+& t &=& 1 \\ 2 x &+& 2 y &+& z &+& t &=& 0 \\ 3 x &+& 2 y &+& 3 z &+& 2 t &=& 3 \\ 6 x &+& 4 y &+& 3 z &+& 2 t &=& 2 \\ \end{array}\right.$$

In [ ]:
# x = 1, y = -2, z = 3, t = -1

6. Rozwiąż układ równań $$\left\{\begin{array}{ccccccccccl} x &-& 2 y && &+& 3 s &+& t &=& 1 \\ 2 x &-& 3 y &+& z &+& 8 s &+& 2 t &=& 3 \\ x &-& 2 y &+& z &+& 3 s &-& t &=& 1 \\ && y&& &+& 3 s &+& 5 t &=& 0 \\ x &-& 2 y && &+& 5 s &+& 8 t &=& -1 \\ \end{array}\right.$$

In [46]:
A = IMatrix([[1,-2,0,3,1,1], [2,-3,1,8,2,3],[1,-2,1,3,-1,1],[0,1,0,3,5,0],[1,-2,0,5,8,-1]], separate=1, var=['x','y','z','s','t'])
show(A)
In [47]:
A.add_multiple_of_row(2, 0, -1)
Out[47]:
\[\left[\begin{array}{rrrrr|r} 1 & -2 & 0 & 3 & 1 & 1\\ 2 & -3 & 1 & 8 & 2 & 3\\ 1 & -2 & 1 & 3 & -1 & 1\\ 0 & 1 & 0 & 3 & 5 & 0\\ 1 & -2 & 0 & 5 & 8 & -1\\ \end{array}\right]\begin{array}{c} \ \\\ \\-1 \cdot w_1\\\ \\\ \end{array}\rightarrow\left[\begin{array}{rrrrr|r} 1 & -2 & 0 & 3 & 1 & 1\\ 2 & -3 & 1 & 8 & 2 & 3\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 1 & 0 & 3 & 5 & 0\\ 1 & -2 & 0 & 5 & 8 & -1\\ \end{array}\right]\]
In [48]:
A.add_multiple_of_row(0, 4, -1)
Out[48]:
\[\left[\begin{array}{rrrrr|r} 1 & -2 & 0 & 3 & 1 & 1\\ 2 & -3 & 1 & 8 & 2 & 3\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 1 & 0 & 3 & 5 & 0\\ 1 & -2 & 0 & 5 & 8 & -1\\ \end{array}\right]\begin{array}{c} -1 \cdot w_5\\\ \\\ \\\ \\\ \end{array}\rightarrow\left[\begin{array}{rrrrr|r} 0 & 0 & 0 & -2 & -7 & 2\\ 2 & -3 & 1 & 8 & 2 & 3\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 1 & 0 & 3 & 5 & 0\\ 1 & -2 & 0 & 5 & 8 & -1\\ \end{array}\right]\]
In [50]:
A.swap_rows(0,4)
Out[50]:
\[\left[\begin{array}{rrrrr|r} 0 & 0 & 0 & -2 & -7 & 2\\ 2 & -3 & 1 & 8 & 2 & 3\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 1 & 0 & 3 & 5 & 0\\ 1 & -2 & 0 & 5 & 8 & -1\\ \end{array}\right]\begin{array}{c} \leftarrow w_5\\\ \\\ \\\ \\\leftarrow w_1 \end{array}\rightarrow\left[\begin{array}{rrrrr|r} 1 & -2 & 0 & 5 & 8 & -1\\ 2 & -3 & 1 & 8 & 2 & 3\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 1 & 0 & 3 & 5 & 0\\ 0 & 0 & 0 & -2 & -7 & 2\\ \end{array}\right]\]
In [51]:
A.add_multiple_of_row(1, 0, -2)
Out[51]:
\[\left[\begin{array}{rrrrr|r} 1 & -2 & 0 & 5 & 8 & -1\\ 2 & -3 & 1 & 8 & 2 & 3\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 1 & 0 & 3 & 5 & 0\\ 0 & 0 & 0 & -2 & -7 & 2\\ \end{array}\right]\begin{array}{c} \ \\-2 \cdot w_1\\\ \\\ \\\ \end{array}\rightarrow\left[\begin{array}{rrrrr|r} 1 & -2 & 0 & 5 & 8 & -1\\ 0 & 1 & 1 & -2 & -14 & 5\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 1 & 0 & 3 & 5 & 0\\ 0 & 0 & 0 & -2 & -7 & 2\\ \end{array}\right]\]
In [52]:
A.add_multiple_of_row(0, 3, 2)
Out[52]:
\[\left[\begin{array}{rrrrr|r} 1 & -2 & 0 & 5 & 8 & -1\\ 0 & 1 & 1 & -2 & -14 & 5\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 1 & 0 & 3 & 5 & 0\\ 0 & 0 & 0 & -2 & -7 & 2\\ \end{array}\right]\begin{array}{c} +2 \cdot w_4\\\ \\\ \\\ \\\ \end{array}\rightarrow\left[\begin{array}{rrrrr|r} 1 & 0 & 0 & 11 & 18 & -1\\ 0 & 1 & 1 & -2 & -14 & 5\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 1 & 0 & 3 & 5 & 0\\ 0 & 0 & 0 & -2 & -7 & 2\\ \end{array}\right]\]
In [53]:
A.add_multiple_of_row(3, 1, -1)
Out[53]:
\[\left[\begin{array}{rrrrr|r} 1 & 0 & 0 & 11 & 18 & -1\\ 0 & 1 & 1 & -2 & -14 & 5\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 1 & 0 & 3 & 5 & 0\\ 0 & 0 & 0 & -2 & -7 & 2\\ \end{array}\right]\begin{array}{c} \ \\\ \\\ \\-1 \cdot w_2\\\ \end{array}\rightarrow\left[\begin{array}{rrrrr|r} 1 & 0 & 0 & 11 & 18 & -1\\ 0 & 1 & 1 & -2 & -14 & 5\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 0 & -1 & 5 & 19 & -5\\ 0 & 0 & 0 & -2 & -7 & 2\\ \end{array}\right]\]
In [54]:
A.add_multiple_of_row(3, 2, 1)
Out[54]:
\[\left[\begin{array}{rrrrr|r} 1 & 0 & 0 & 11 & 18 & -1\\ 0 & 1 & 1 & -2 & -14 & 5\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 0 & -1 & 5 & 19 & -5\\ 0 & 0 & 0 & -2 & -7 & 2\\ \end{array}\right]\begin{array}{c} \ \\\ \\\ \\+1 \cdot w_3\\\ \end{array}\rightarrow\left[\begin{array}{rrrrr|r} 1 & 0 & 0 & 11 & 18 & -1\\ 0 & 1 & 1 & -2 & -14 & 5\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 0 & 0 & 5 & 17 & -5\\ 0 & 0 & 0 & -2 & -7 & 2\\ \end{array}\right]\]
In [55]:
A.add_multiple_of_row(3, 4, 2)
Out[55]:
\[\left[\begin{array}{rrrrr|r} 1 & 0 & 0 & 11 & 18 & -1\\ 0 & 1 & 1 & -2 & -14 & 5\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 0 & 0 & 5 & 17 & -5\\ 0 & 0 & 0 & -2 & -7 & 2\\ \end{array}\right]\begin{array}{c} \ \\\ \\\ \\+2 \cdot w_5\\\ \end{array}\rightarrow\left[\begin{array}{rrrrr|r} 1 & 0 & 0 & 11 & 18 & -1\\ 0 & 1 & 1 & -2 & -14 & 5\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 0 & 0 & 1 & 3 & -1\\ 0 & 0 & 0 & -2 & -7 & 2\\ \end{array}\right]\]
In [56]:
A.add_multiple_of_row(4, 3, 2)
Out[56]:
\[\left[\begin{array}{rrrrr|r} 1 & 0 & 0 & 11 & 18 & -1\\ 0 & 1 & 1 & -2 & -14 & 5\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 0 & 0 & 1 & 3 & -1\\ 0 & 0 & 0 & -2 & -7 & 2\\ \end{array}\right]\begin{array}{c} \ \\\ \\\ \\\ \\+2 \cdot w_4 \end{array}\rightarrow\left[\begin{array}{rrrrr|r} 1 & 0 & 0 & 11 & 18 & -1\\ 0 & 1 & 1 & -2 & -14 & 5\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 0 & 0 & 1 & 3 & -1\\ 0 & 0 & 0 & 0 & -1 & 0\\ \end{array}\right]\]
In [57]:
A.rescale_row(4,-1)
Out[57]:
\[\left[\begin{array}{rrrrr|r} 1 & 0 & 0 & 11 & 18 & -1\\ 0 & 1 & 1 & -2 & -14 & 5\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 0 & 0 & 1 & 3 & -1\\ 0 & 0 & 0 & 0 & -1 & 0\\ \end{array}\right]\begin{array}{c} \ \\\ \\\ \\\ \\\cdot -1 \end{array}\rightarrow\left[\begin{array}{rrrrr|r} 1 & 0 & 0 & 11 & 18 & -1\\ 0 & 1 & 1 & -2 & -14 & 5\\ 0 & 0 & 1 & 0 & -2 & 0\\ 0 & 0 & 0 & 1 & 3 & -1\\ 0 & 0 & 0 & 0 & 1 & 0\\ \end{array}\right]\]

7. Rozwiąż układ równań $$\left\{\begin{array}{ccccccccl} x &+& 2 y &+& z &+& t &=& 7 \\ 2 x &-& y &-& z &+& 4 t &=& 2 \\ 5 x &+& 5 y &+& 2 z &+& 7 t &=& 1 \\ \end{array}\right.$$

8. Rozwiąż układ równań $$\left\{\begin{array}{ccccccccl} x &+& 2 y &+& 3 z &+& t &=& 1 \\ 2 x &+& 4 y &-& z &+& 2 t &=& 2 \\ 3 x &+& 6 y &+& 10 z &+& 3 t &=& 3 \\ x&+& y&+& z&+& t &=& 0 \\ \end{array}\right.$$

9. Rozwiąż układ równań $$\left[\begin{array}{rrrr|r} 3 & 2 & 1 & -1 & 0\\ 5 & -1 & 1 & 2 & -4\\ 7 & 8 & 1 & -7 & 6\\ 1 & -1 & 1 & 2 & 4\\ \end{array}\right]$$

10. Rozwiąż układ równań $$\left[\begin{array}{rrrrr|r} 1 & -3 & 1 & -2 & 1 & -5\\ 2 & -6 & 0 & -4 & 1 & -10\\ 0 & 0 & 2 & 0 & 1 & 0\\ -2 & 6 & 2 & 4 & 0 & 10\\ -2 & 6 & 4 & 4 & 1 & 10\\ -1 & 3 & 1 & 2 & 0 & 5\\ \end{array}\right]$$

11. W zależności od wartości parametru $a$ powiedzieć, czy układ ma jedno rozwiązanie / ma wiele rozwiązań / jest sprzeczny. $$ \left\{\begin{array}{ccccccl} a x&+& y&+& z&=&a - 1 \\ x&+& y&+&a z &=& 1 \\ x&+&a y&+& z&=&-a + 1 \\ \end{array}\right.$$

12. Niech $a, b, c$ będą trzema różnymi liczbami rzeczywistymi. Znaleźć wielomian kwadratowy $$W(x) = \lambda x^2 + \mu x + \nu$$ taki, że $$W(a) = 7, W(b) = 4 \text{ i } W(c) = 9.$$