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Show there exists an inner product on V such that V's basis is an orthonormal basis.

If i let the inner product be $\langle v_i,v_j \rangle$ such that

$\langle v_i,v_j \rangle=1$ if $i=j$ and $\langle v_i,v_j \rangle=0$ if $i \neq j$ then this makes the basis orthonormal, correct?

Now I am asked:

"show that the inner product is unique"

Do I start with "suppose $[v_i,v_j]=1$ if $i=j$ and $\langle v_i,v_j \rangle=0$ if $i \neq j$?

Did i do the first part incorrectly? How can I show an inner product is unique?

  • 0
    Show that $\langle a,b\rangle$ has only one possible value for any two vectors $a$ and $b$ – 2017-01-29

1 Answers 1

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What you're done is correct so far. Because of linearity, it suffices to define an inner product in $V$ to pick up a basis $\{v_1,..., v_n\}$ and define $\langle v_i, v_j\rangle$, for $i, j \in \{1,...,n\}$.