So the question is:
Which of the following sets of vectors form an orthonormal basis for $\mathbb{R}^2$
$(a) \{(1,0)^T, (0,1)^T\}$
$(b) \{(\frac{3}{4},\frac{4}{5})^T,(\frac{5}{13},\frac{12}{13})^T\}$
I know that a is a basis and b is not. The thing is, I just don't know why. I know the definition of orthonormal is if two vectors are perpendicular and of unit length. But I don't understand how to prove or even find for that matter, the orthonormal basis of a space. Any ideas? Thanks
Best Answer
Two vectors $x,y \in \mathbb{R}^n$ are orthogonal if their dot product equals zero; that is $$ x \cdot y = x_1 y_1 + \ldots + x_n y_n = 0,$$ where $x = (x_1, \ldots, x_n)$, $y = (y_1, \ldots, y_n)$. A vector $x$ is of unit length if $x \cdot x = 1$. A set is called orthonormal if every vector has unit length and any two different vectors are orthogonal. We also know the following:
So we can just compute the dot products between all the vectors in your examples, and check whether they equal 0 when dotting two different vectors, and equals 1 when dotting a vector with itself. Then, using the theorem, you'll automatically know whether it's a orthonormal basis or not.