Linear independence

**bigdoggy** · April 16th 2009, 02:43 PM

Given two vectors x,y where length of x = 1, length of y = 5 and their scalar product = 3. How could I show the ordered pair {x,y} is a basis?

**Andres Perez** · April 16th 2009, 05:48 PM

Suppose that they are dependent, so there exists $\alpha \in \mathbb{R}$ such that

$X=\alpha Y$

put $Y=(y_1,y_2)$ , then $X=(\alpha y_1,\alpha y_2)$

Now,

$3=\left< X,Y\right>=\alpha (y_1^2+y_2^2)=\alpha \Vert Y \Vert ^2=25\alpha$

hence $\alpha =\frac{3}{25}$ , thus

$\Vert X \Vert=\sqrt{(\alpha y_1)^2+(\alpha y_2)^2}=\alpha \Vert Y \Vert =\frac{3}{5}$

a contradiction. Therefore, they are independent

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