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make much sense to me (I mean, even basic properties). I

read something about the dot product, then they started getting into

<x+y, z> is <x,z>+<y,z> what is the purpose of doing this?

I'm almost completely clueless about inner products.

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- Thread starter evilpostingmong
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make much sense to me (I mean, even basic properties). I

read something about the dot product, then they started getting into

<x+y, z> is <x,z>+<y,z> what is the purpose of doing this?

I'm almost completely clueless about inner products.

- #2

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I'm not a fan of that notation. Are you familiar with upper and lower indices?

- #3

tiny-tim

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It's the same rule as for dot-products of ordinary 3D vectors …

(

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Don't mean to sound pushy, just in a desperate mood.

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HallsofIvy

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Yes, it is important.

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Hurkyl

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It's hard to give a good answer without more context (e.g. what is your background? What are you actually studying?), but I'll make a try anyways.

make much sense to me (I mean, even basic properties). I

read something about the dot product, then they started getting into

<x+y, z> is <x,z>+<y,z> what is the purpose of doing this?

I'm almost completely clueless about inner products.

The big overarching incredible point about algebra is that it's not just something you do with numbers. You do also do it with vectors, matrices, sets, geometric shapes -- pretty much anything you would ever want to study can be studied with

Linear algebra has a special role, because it is the simplest kind of algebra, and something we understand really,

In order to do algebra, we need to know how to manipulate equations. The reason you learn the law [itex]\langle \vec{x}+\vec{y}, \vec{z} \rangle = \langle \vec{x}, \vec{z}\rangle + \langle \vec{y}, \vec{z} \rangle[/itex] for manipulating vectors is exactly the same as the reason you learn the law [itex]a(b+c) = ab + ac[/itex] for manipulating numbers.

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I couldn't even figure out what exactly was confusing..I know) is

that the inner product is supposed to be a scalar but when

computing it <x+y, z>=<x,z>+<y,z> <x,z> and <y,z> look like

bases containing vectors, not scalars. I guess that's why

Phrak doesn't like this notation. Those don't look like scalars, so I

don't really know what x or y or z are. Unless that "," means multiplication.

Oh btw to answer Hurky's question, as far as linear algebra is concerned.

I know matrix arithmetic, vector spaces, linear transformations, and "eigenstuff".

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Hurkyl

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If inner products are scalar-valued...the inner product is supposed to be a scalar ... Those don't look like scalars,

And <x,z> denotes an inner product of x with z...

Then <x,z> is a scalar.

I'm not sure what you mean by "bases containing vectors".

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It's alright, you know how when you want to write out a basisIf inner products are scalar-valued...

And <x,z> denotes an inner product of x with z...

Then <x,z> is a scalar.

I'm not sure what you mean by "bases containing vectors".

for a vector space of dim n you'd put <v1...vn>

Oh hold on x+y and z are vectors but you performed the cross

product to get xy and xz, which are scalars.

But why couldn't they write it as <x*y> or <x*z> isn't that more obvious?

I'm not blaming you or anyone else on this forum, so don't worry about that.

Wait, I thought of something. Take vector u to be a row [1 2] and v to be a column [3 4]

so the result is [4 6] after multilpication so I end up with (for <u, v>) u is the scalar 4 and v

is the scalar 6.

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- #11

Hurkyl

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There aren't that many symbols useful for a binary operation. Asterisks (i.e. '*') are annoying to draw by hand, if you have to do it a lot. The dot (i.e. '[itex]\cdot[/itex]') is good, but people often reserve that for one specific inner product: the dot product on

It's awkward to write the dot product as if it were ordinary multiplication, because 'ordinary multiplication' notation is already overused in linear algebra. e.g. if

* Scalar multiplication: [itex]r \vec{v}[/itex]

* Scalar multiplication: [itex]r A[/itex]

* Matrix-matrix product: [itex]A B[/itex]

* Matrix-vector product: [itex]A v[/itex]

* Applying a transformation: [itex]T v[/itex]

I think it should be preferable to let ordinary multiplication denote

Another advantage to some sort of bracket notation (e.g. [itex]\{ a, b \}[/itex] or [itex]\langle a, b \rangle[/itex] or [itex][ a, b ][/itex]) is that it can be more easily annotated. If you're working with two different inner products, we can label one

How did the notation actually originate? I don't know. I can speculate, though: I bet the it was originally written as an ordinary binary function: e.g.

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HallsofIvy

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I hope you are not confusing <x, y> with {x, y}! If so, you need to have your vision checked. x, y, and z are vectors here. So is x+ y. <x+y, z>, <x, z>, and <y, z> are all inner products of vectors and so are scalars. <x, z>+ <y, z> is a sum of scalars, equal to the scalar <x+ y, z>.Thanks for the input. But what confuses me (I was so confused that

I couldn't even figure out what exactly was confusing..I know) is

that the inner product is supposed to be a scalar but when

computing it <x+y, z>=<x,z>+<y,z> <x,z> and <y,z> look like

bases containing vectors, not scalars. I guess that's why

Phrak doesn't like this notation.

I presume that the first statement in the definition of "inner product" in your textbook is that the inner product "is a function from VxV to the underlying field": that is, that the inner product, symbolized by < , >, takes two vectors, say x and y, and changes them to theThose don't look like scalars, so I

don't really know what x or y or z are. Unless that "," means multiplication.

Oh btw to answer Hurky's question, as far as linear algebra is concerned.

I know matrix arithmetic, vector spaces, linear transformations, and "eigenstuff".

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- #13

tiny-tim

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How did the notation actually originate? … it got shortened to(x,y). But it can be confusing to use parentheses, so they switched to angle brackets [itex]\langle x, y \rangle[/itex].

Hi Hurkyl!

I always thought

or did they already exist, and he only gave them the humorous name?

- #14

Hurkyl

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This is how I thought history went. (And also other bracket operators pre-existed, like the Poisson bracket) But I don't have great confidence in my knowledge of such things. Your version is entirely plausible.or did they already exist, and he only gave them the humorous name?

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I hope you are not confusing <x, y> with {x, y}! If so, you need to have your vision checked. x, y, and z are vectors here. So is x+ y. <x+y, z>, <x, z>, and <y, z> are all inner products of vectors and so are scalars. <x, z>+ <y, z> is a sum of scalars, equal to the scalar <x+ y, z>.

I presume that the first statement in the definition of "inner product" in your textbook is that the inner product "is a function from VxV to the underlying field": that is, that the inner product, symbolized by < , >, takes two vectors, say x and y, and changes them to thescalar<x, y>. From that <x, z>, <y, z>, and <x+ y, z> certainly should "look like scalars"! Anything of the form <u, v>isa scalar.

Oh, its ok I have a book that uses <> for bases as well, but I guess it was a stupid

idea for the book to use <> for bases since it would confuse students who are

learning linear algebra. But you have shown me what they are normally used for,

thank you very much! And thanks Hurkyl for solving my , and * dilemma!

Thank you all for responding!

Oh btw Hurkyl, when you say "normal multiplication" you mean take two vectors and multiply them

like this uv as opposed to dot product, which is u

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