Does the Given Condition Prove Vector Belongs to Span of Orthonormal Set?

[nhrock3]

there is an orthonormal group {u1,..,uk} in R^n

there is vector v which belongs to R^n

prove that if

||v||^2=(v*u1)^2 +..+(v*u_k)^2

then v belongs to the sp{u1..uk}

*-is dot product



how i tried to solve it:

i expanded the orthonormal group {u1,..,uk} to

the orthonormal group {u1,..,uk,..un}

then v is its combination

v=a1u1+a2u2+..anun

i put v in the given formula

||v||^2=((a1u1+a2u2+..anun)*u1)^2 +..+((a1u1+a2u2+..anun)*u_k)^2

=(a1u1^2)^2 +..(akuk^2)^2=a1^2+..a^k^2

u1..uk are orthonormal

so u1^2=1.. uk^2=1

what now?

 

[micromass]

Hi nhrock3!

Can you write out $\|v\|^2$? You can use

$$\|v\|^2=<v,v>=<a_1u_1+...+a_nu_n,a_1u_1+...+a_nu_n>$$