# Linear functional

• May 9th 2009, 08:27 AM
frater_cp
Linear functional
Let X be n-dimensional vector space. Z a proper substance of X.
Let x_0 be element of X - Z (complement of Z).

Show there is a linear functional f on X such that:

f(x_0)=1 and f(x)=0 for all x element of Z.
• May 9th 2009, 05:43 PM
hkito
Do you know about inner product?
• May 10th 2009, 08:05 AM
HallsofIvy
Construct a basis for Z, say $\{u_1, u_2, ..., u_n\}$. Extend that to a basis for X: $\{u_1, u_2, ..., u_n, v_1, v_2, ..., v_m\}$.

Define $f(u_i)= 0$, $f(v_i)= 1$, and extend to all of X by "linearity".
• May 11th 2009, 11:29 AM
frater_cp