i'm having a problem with the following problem. i have 2 sentences and i neech to check if they're both right, both wrong or only one is right.
1)let V be an inner product space, $B = {u_1,...,u_n}$ is a basis of V and we have scalars $j_1,...,j_n$ so that ${||\sum_{k=1}^n j_ku_k||}^2= \sum_{k=1}^n |j_k|^2$.
2)$U=sp({1,x,x^2})$ sub space of $R_5[x]$ with the integral inner product in [0,1]. the orthogonal projection of x^3 on U is $(x^3,1)1+(x^3,x)+(x^3,x^2)x^2$
and a small question that i think might not be worthy of a thread of its own. how can i be sure that a matrix is completely(definite) positive? for instance in the case of $\begin{pmatrix}1&4\\ 4&6\end{pmatrix}$?
what i tried:
1)true.using the homogenity property of the normal (i.e j is a scalar, ||: V->R, so ||jv||=||v||*|j|), if $u_k$ is 1, then ||v|| is a normal.
2)true.since the length is 1, then $(x^3,1)1+(x^3,x)+(x^3,x^2)x^2$ can be concluded as a orthogonal projection of $x^3$ on U
3)regarding the completely positive matrix. i need a matrix B, so that if $A=\begin{pmatrix}1&4\\ 4&6\end{pmatrix}$, then $A=BB^T$, but i don't know how to find B or how to determine if it's completely(definite) positive matrix.
thank you very much for helping. did my best to elabroate.