Page:Linear Algebra (1882) Tevfik.djvu/23

 will give the same result. In the case in which the lines to be added are in the same direction, this operation is reduced to the addition of Numerical Algebra.

16. Is the operation of finding one of two lines, when the other and their sum are given. To subtract $$A D$$ from $$A B$$, or to find a line which added to $$A D$$ will produce the line $$A B$$, it is evident that if we trace from the point $$B$$ a line equal to $$D A$$, we shall have the line $$A N$$ which added to $$A D$$ will produce $$A B$$:

$A N = A B - A D$

and

$A N + A D = A B$.

A few propositions on the employment of lines in Algebraic operations.

17. In the case that we have

$A B + B D + D H = A H$

we shall have also

$n \times A B + n \times B D + n \times D H = n \times A H$,

in designating by n an abstract number. And if we have

$n \times AB + n \times B D + n \times D H = n \times A H$

we shall have

$A B + B D + D H = A H$.

In tracing the expressions $$A B + B D + D H$$, and $$n \times A B + n \times B D + n \times D H$$, the truth of the proposition will be manifest. Thus if we designate the lines $$A B, B D, D H$$, and $$H L$$ by the Greek letters $$\alpha, \beta, \gamma, \delta$$, and if we have, for exemple,

$28\alpha + 21\beta + 7\gamma + 49\delta = 14\omega$.

We shall have also as in Numerical Algebra

$7 (4\alpha + 3\beta + \gamma + 7\delta) = 7. 2\omega$

or

$4\alpha + 3\beta + \gamma + 7\delta = 2\omega$.

18. If the lines $$\alpha, \beta$$ have not the same direction, and we designate by $$a$$ and $$b$$ two abstract numbers, the lines $$a\alpha, b\beta$$ cannot neutralize each other in Algebraic expressions. Therefore if as a result of some operation we have,

$a\alpha + b\beta = 0$

we shall conclude that $$a = 0, b = 0$$.