Complete Example AOS by verifying that the four remaining inner products are zero.

Verify that the set $T$ created in Example GSTV by the Gram-Schmidt Procedure is an orthogonal set.

Suppose that $\set{\vect{u},\,\vect{v},\,\vect{w}}\subseteq\complex{n}$ is an orthonormal set. Prove that $\vect{u}+\vect{v}$ is not orthogonal to $\vect{v}+\vect{w}$.

Prove part 2 of the conclusion of Theorem IPVA.

Prove part 2 of the conclusion of Theorem IPSM.

Suppose that $\vect{u},\,\vect{v},\,\vect{w}\in\complex{n}$, $\alpha,\,\beta\in\complexes$ and $\vect{u}$ is orthogonal to both $\vect{v}$ and $\vect{w}$. Prove that $\vect{u}$ is orthogonal to $\alpha\vect{v}+\beta\vect{w}$.

Suppose that the set $S$ in the hypothesis of Theorem GSP is not just linearly independent, but is also orthogonal. Prove that the set $T$ created by the Gram-Schmidt procedure is equal to $S$. (Note that we are getting a stronger conclusion than $\spn{T}=\spn{S}$ — the conclusion is that $T=S$.) In other words, it is pointless to apply the Gram-Schmidt procedure to a set that is already orthogonal.

Suppose that the set $S$ is linearly independent. Apply the Gram-Schmidt procedure (Theorem GSP) twice, creating first the linearly independent set $T_1$ from $S$, and then creating $T_2$ from $T_1$. As a consequence of Exercise O.T30, prove that $T_1=T_2$. In other words, it is pointless to apply the Gram-Schmidt procedure twice.