Understanding Orthogonality in Inner Product Spaces

In summary, we learned that when $(x,x)=0$, we can conclude that $x=0$. However, when we have $(x,y)=0$, we cannot conclude that $x=0$ AND $y=0$, but we can say that $x$ and $y$ are perpendicular or one of them is zero. Additionally, we saw an example of this with $(\hat \imath, \hat \jmath) = 0$. Finally, we received clarification that this was helpful from (Sun).
  • #1
mathmari
Gold Member
MHB
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Hey! :eek:

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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  • #2
mathmari said:
Hey! :eek:

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)
 
  • #3
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)
 

FAQ: Understanding Orthogonality in Inner Product Spaces

What is the "Identity of inner product"?

The identity of inner product refers to a mathematical property of an inner product space, where the inner product of a vector with itself results in the length of the vector squared. This identity is also known as the Pythagorean theorem in Euclidean spaces.

How is the identity of inner product used in mathematics?

The identity of inner product is a fundamental property in linear algebra and functional analysis. It is used to define the norm and distance in an inner product space, and it plays a crucial role in proving the convergence of sequences and series in these spaces. It also has applications in physics, computer science, and other fields.

What are the conditions for an inner product space to have the identity of inner product?

An inner product space must satisfy certain conditions for the identity of inner product to hold. These conditions include linearity in the first argument, conjugate symmetry, and positive definiteness. In other words, the inner product of a vector with itself must be a positive real number.

Can the identity of inner product be generalized to other types of spaces?

Yes, the identity of inner product can be generalized to other types of spaces, such as complex inner product spaces and inner product spaces over other fields. However, the specific conditions for the identity to hold may differ in these cases.

Why is the identity of inner product important?

The identity of inner product is important because it allows us to define the length and angle of vectors in an inner product space. It also provides a way to measure the distance between vectors and to define important concepts, such as orthogonality and orthonormality. These concepts are essential in many areas of mathematics, physics, and engineering.

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