Relation between definition of Subspaces

[tardy narwhal]

Is there a relation between this definition of a null space and the null space (kernel) of a linear map/matrix?

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[lusty birch]

Typically $\langle v,v\rangle=0$ implies that $v=0$ for an inner product. So I think when they say "null space" they literally mean the vector space containing only the zero vector.

[ocean wadi]

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[tardy narwhal]

I don't understand how $\langle v, v \rangle = 0 \implies v=0$. What do you mean "typically"?

[ocean wadi]

omeganebula

[tardy narwhal]

Like what's stopping me from making the inner product for any two vectors zero by definition and calling every vector space a null space

[vast egret]

By definition, an inner product has the property <v|v>=0 =>v=0 to be able to define a norm

[lusty birch]

Usually you need these three properties

So just check the definition in your notes and see if you get the implication as definition or not

[lusty birch]

Nothing is stopping you from doing this. The implication of doing this is that your vector space turns into just the zero vector by doing this.

[tardy narwhal]

In the book the positive definite property is not required to define a scalar product

Is there a difference between inner product and scalar product?

[grim leaf]

the scalar product is an inner product but not the other way around

Scalar products are an euclidean thing, inner products generalise them to more abstract spaces

[tardy narwhal]

I don't see how they generalise, what's the difference in their definitions

[tardy narwhal]

Hermitian products aren't inner products according to this

[lusty birch]

Well let's work this is a different order. What's the definition you have been given?

[lusty birch]

The one I posted was for real inner product spaces

[tardy narwhal]

One sec

[tardy narwhal]

Oh

So that's the same as the so called scalar product?

ig the author shouldve mentioned non-degenerate in the original text

[lusty birch]

Yeah that would have helped with the confusion for sure

So it's the non degenerate property we're all letting you know must be satisfied in order for the product to be somewhat useful

[tardy narwhal]

Alright makes sense

Thanks!

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