/sci/ - Science & Math

>>9014821
For simplicity, let [math]x, y\in\mathbb{R}^n, n>1[/math], and consider the ordinary inner product. Now, [math]x, y[/math] are orthogonal iff [math]x\cdot y=\sum\limits_{k=1}^n x_ky_k=0[/math]. This can then be generalized to any inner product [math]\langle\cdot, \cdot\rangle[/math] of Euclidean spaces to get generalized orthogonality: [math]x, y[/math] are orthogonal iff [math]\langle x, y\rangle=0[/math]. But notice that we didn't need any properties of Euclidean spaces there, and so we can replace [math]\mathbb{R}^n[/math] by any vector space such that one can define an inner product in.

Notice that you don't have [math]\langle x, \langle y, z\rangle\rangle[/math] even defined, since the inputs are vectors and the output is a scalar.

>>8508462
Yes, the amount of applications is gigantic. It is good to find stuff intuitive, as that makes things a lot easier to handle. And, also, it is a good thing that you find analysis intuitive, considering that you are going to be an engineer.

>>8508466
That is why you should record your ideas somehow. I usually write them down, and my home looks like I was some schizophrenic as I have my little reminder papers taped all over the place.