### Vector and Matrix

Exercise Express the vectors **b** =
(4, 6), **c** = (1, 1), **e**_{1} = (1, 0), **e**_{2}
= (0, 1) as linear combinations of the vectors **u** = (1, 2), **v**
= (3, 4), **w** = (5, 6). Can you express as linear combinations of
the vectors **u**, **v** only?

Answer We would like to write **b**,
**c** , **e**_{1}, **e**_{2} in
the form `x`_{1}(1, 2) + `x`_{2}(3, 4) + `x`_{3}(5,
6). From the previous exercise,
we have

`x`_{1}[ |
1 |
] + `x`_{2}[ |
3 |
] + `x`_{3}[ |
5 |
] = [ |
`x`_{1} + 3`x`_{2} + 5`x`_{3} |
] |

2 |
4 |
6 |
2`x`_{1} + 4`x`_{2} + 6`x`_{3} |

Therefore the problem is to solve the system of equations

`x`_{1} |
+ 3`x`_{2} |
+ 5`x`_{3} |
= |
`b`_{1} |

2`x`_{1} |
+ 4`x`_{2} |
+ 6`x`_{3} |
= |
`b`_{2} |

For `b`_{1} = 4, `b`_{2} = 6 on the right side, we
find `x`_{1} = 1, `x`_{2} = 1 is one solution. Therefore
** **`b` = **u** + **v**. In fact, the solution
is not unique. For example, `x`_{1} = 2, `x`_{2} =
-1, `x`_{3} = 1 is another solution, from which we get ** **`b`
= 2**u** - **v** + **w**.

Incidentally, we have already expressed **b** as a linear combination
of **u** and **v**: **b** = **u** +
**v**. Such expressions are equivalent to solutions of

`x`_{1} |
+ 3`x`_{2} |
= |
`b`_{1} |

2`x`_{1} |
+ 4`x`_{2} |
= |
`b`_{2} |

We do the similar computation for the other vectors and find **c**
= -1/2 **u** + 1/2 **v**, **e**_{1} =
-2**u** + **v**, **e**_{2} = 3/2 **u**
- 1/2 **v**.