# inner product spaces

• Oct 18th 2008, 10:24 PM
bamby
inner product spaces
Can you please, help me with this
Suppose V and W are finite dimensional inner product spaces with orthogonal bases B1 and B2, respectively. Let T: V->W is linear, so we know that T*: V->W (linear) exists and is unique. Prove that [T*]_B1,B2 is the conjugate transpose of [T*]_B2,B1.
Thank you!
• Oct 19th 2008, 07:19 AM
Opalg
Quote:

Originally Posted by bamby
Suppose V and W are finite dimensional inner product spaces with orthogonal bases B1 and B2, respectively. Let T: V->W is linear, so we know that T*: V->W (linear) exists and is unique. Prove that [T]_B1,B2 is the conjugate transpose of [T*]_B2,B1.

Suppose that B_1 consists of vectors $\displaystyle e_i$, and B_2 consists of vectors $\displaystyle f_j$. Then $\displaystyle \langle Te_i,f_j\rangle = \langle e_i,T^*f_j\rangle = \overline{\langle T^*f_j,e_i\rangle}$. In other words, the (j,i)-element of $\displaystyle [T^*]_{B_2,B_1}$ is the complex conjugate of the (i,j)-element of $\displaystyle [T]_{B_1,B_2}$.