Could anyone elaborate on why having an orthonormal basis would make (u_1^2 + ... + u_n^2) the length of u?
bangyanc
I don't have a complete answer but I think this has to do with how inner product is defined.
elenagong
I believe it is just the definition of norm?
motoole2
There's a mistake with the slide: one must take the square root of (u_1^2 + ... + u_n^2) to get the norm/length of vector u.
The question still remains though: why does the Euclidean norm equal to the length of u only makes sense if the corresponding basis vectors are unit length and orthogonal (i.e., orthonormal)?
Let's define the vector v = u_1 * e_1 + u_2 * e_2, where e_1 and e_2 are our basis vectors and (u_1, u_2) is the corresponding vector. The length of v is given by its norm: |v| = sqrt(<u_1 * e_1 + u_2 * e_2, u_1 * e_1 + u_2 * e_2>).
If we expand this out, we get the following: |v| = sqrt(u_1^2 * <e_1, e_1> + u_2^2 <e_2, e_2> + u_1 u_2 * <e_1, e_2> + u_2 u_1 * <e_2, e_1>). If our basis were orthonormal, then this reduces nicely down to the expression for the Euclidean norm, i.e., setting <e_1, e_2> = 0 and <e_1, e_1> = 1. This is not the case otherwise.
If it's still unclear, try drawing this out to see what happens when e_1 and e_2 are not orthonormal!
Could anyone elaborate on why having an orthonormal basis would make (u_1^2 + ... + u_n^2) the length of u?
I don't have a complete answer but I think this has to do with how inner product is defined.
I believe it is just the definition of norm?
There's a mistake with the slide: one must take the square root of
(u_1^2 + ... + u_n^2)to get the norm/length of vectoru.The question still remains though: why does the Euclidean norm equal to the length of
uonly makes sense if the corresponding basis vectors are unit length and orthogonal (i.e., orthonormal)?Let's define the vector
v = u_1 * e_1 + u_2 * e_2, wheree_1ande_2are our basis vectors and(u_1, u_2)is the corresponding vector. The length ofvis given by its norm:|v| = sqrt(<u_1 * e_1 + u_2 * e_2, u_1 * e_1 + u_2 * e_2>).If we expand this out, we get the following:
|v| = sqrt(u_1^2 * <e_1, e_1> + u_2^2 <e_2, e_2> + u_1 u_2 * <e_1, e_2> + u_2 u_1 * <e_2, e_1>). If our basis were orthonormal, then this reduces nicely down to the expression for the Euclidean norm, i.e., setting <e_1, e_2> = 0 and <e_1, e_1> = 1. This is not the case otherwise.If it's still unclear, try drawing this out to see what happens when e_1 and e_2 are not orthonormal!