$\ell_2$ norm vs inner product

Question

On a homework sheet I am working on, one of the problems has us calculate the inner product of a vector with itself. The vector here is $u=(0.1^2, 0.1^2, 0.1^2, \dots, 0.1^2)$, and the number of elements in this vector is $10^6$. The first part of the question is to calculate $u^Tu$ which is equal to $10^4$. Now my confusion begins with the next question, we are asked to calculate $$\sqrt{\sum_{i=1}^{10^6}|u_i|^2} $$ which I have calculated to be $100$. The wording of the question seems to be implying that the two quantities should be the same, am I incorrect or is the wording a bit misleading?

Answer 1

The quantity $u^Tu$ is the standard inner product $\langle u,u\rangle$, which equals $\Vert u\Vert^2$, i.e. the squared norm. The other quantity that you have $$\sqrt{\sum_{i=1}^{10^6} |u_i|^2}$$ is equal to the norm $\Vert u\Vert$ exactly. If you remove the big square root symbol in the above quantity, then these two numbers are the same.