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I have an exercise that I am trying to decipher, but as I have never seen this notation before I do not know how to read it. The problem states:

The vectors $x_1=(1,1,1),x_2=(1,1,-1)$ and $x_3=(1,-1,-1)$ form a basis of $\mathbb{C}^3$. If ${y_1,y_2,y_3}$ is the dual basis, and if $x=(0,1,0)$, find $[x,y_1],[x,y_2],[x,y_3]$.

Can someone help me understand what this last part $[x,y_i]$ means?

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  • $\begingroup$ Welcome to Mathematics Stack Exchange. It could be the inner product $\endgroup$ – J. W. Tanner May 31 at 18:56
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Your exercise is from Halmos: Finite-Dimensional Vector Spaces.

If $V$ is a vector space, $y \in V'$ (the dual space of $V)$ and $x \in V$, the notation $[x, y]$ stands for the scalar $y(x)$, which one obtains when one inserts $x$ into $y$. It is explained in the text why this notation is used.

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  • $\begingroup$ Right, that notation, $y(x)$, is what I am used to. That solves it. $\endgroup$ – Jake May 31 at 19:06

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