MHB Understanding Orthogonality in Inner Product Spaces

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In inner product spaces, the condition \( (x,x)=0 \) implies that \( x=0 \). However, if \( (x,y)=0 \), it only indicates that \( x \) and \( y \) are orthogonal, meaning they are perpendicular, or that at least one of them is zero. It does not necessarily mean that both \( x \) and \( y \) must be zero. This distinction is important in understanding the properties of inner product spaces. Overall, orthogonality does not imply that both vectors are null.
mathmari
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Hey! :o

We know that:
$$(x,x)=0 \Rightarrow x=0$$

When we have $\displaystyle{(x,y)=0}$, do we conclude that $\displaystyle{x=0 \text{ AND } y=0}$. Or is this wrong? (Wondering)
 
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mathmari said:
Hey! :o

We know that:
$$(x,x)=0 \Rightarrow x=0$$

When we have $\displaystyle{(x,y)=0}$, do we conclude that $\displaystyle{x=0 \text{ AND } y=0}$. Or is this wrong? (Wondering)

Hi hi! (Happy)

In this case we can only say that $x$ and $y$ are perpendicular, or one of them is zero.
Note that $(\hat \imath, \hat \jmath) = 0$. (Wasntme)
 
I like Serena said:
Hi hi! (Happy)

In this case we can only say that $x$ and $y$ are perpendicular, or one of them is zero.
Note that $(\hat \imath, \hat \jmath) = 0$. (Wasntme)

I see! Thanks a lot! (Sun)
 
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