Linear Algebra define scalar products

[Physicstcd14]

Homework Statement

How do you know if say [(x_1,y_1),(x_2,y_2)] = x_1x_2 + 7y_1y_2 ? or any other equation?

Homework Equations

The Attempt at a Solution

 

[haruspex]

Assuming the 7 is a typo, it's a matter of definition.

 

[Stephen Tashi]

Your question isn't clear. Try rephrasing it. Specify the topic you are talking about. Are you asking something about "inner product" or "dot product"?

 

[Physicstcd14]

The question I was given on my Linear Algebra home assignment is as follows
Witch of the following formulas define scalar products on R^2? Explain your answer.
(a) ((x_1,y_1),(x_2,y_2)) = x_1y_2 + x_2y_1

(b) ((x_1,y_1),(x_2,y_2)) = x_1x_2 + 7y_1y_2

(c) ((x_1,y_1),(x_2,y_2)) = x_1x_2 + x_1y_2 + x_2y_1 + y_1y_2

I know how to show its positive def. and I know how to show it is symmetric. I Want to know how to show its bilinear.

Sorry I left out bilinear in my original question :/

 

[Stephen Tashi]

The question I was given on my Linear Algebra home assignment is as follows
Witch of the following formulas define scalar products on R^2? Explain your answer.
(a) ((x_1,y_1),(x_2,y_2)) = x_1y_2 + x_2y_1

(b) ((x_1,y_1),(x_2,y_2)) = x_1x_2 + 7y_1y_2

(c) ((x_1,y_1),(x_2,y_2)) = x_1x_2 + x_1y_2 + x_2y_1 + y_1y_2

I know how to show its positive def. and I know how to show it is symmetric. I Want to know how to show its bilinear.

Sorry I left out bilinear in my original question :/

What's the definition of "bilinear" in your course materials? Is it http://en.wikipedia.org/wiki/Bilinear_form ?
Then show
( (u_x,u_y) + (v_x,v_y), (w_x,w_y) ) = ((u_x,u_y),(w_x,w_y)) + ((v_x,v_y)(w_x,w_y))
and
<br /> ( (u_x,u_y), (v_x,v_y)+ (w_x,w_y) ) = ((u_x,u_y),(v_x,v_y)) + ((u_x,u_y)(w_x,w_y))
and
(\lambda(u_x,u_y), (v_x,v_y)) = ((u_x,u_y),\lambda(v_x,v_y)) = \lambda( (u_x,u_y),(v_x,v_y))

 

[Physicstcd14]

Oh right okay. Thanks

 

[Ray Vickson]

What's the definition of "bilinear" in your course materials? Is it http://en.wikipedia.org/wiki/Bilinear_form ?
Then show
( (u_x,u_y) + (v_x,v_y), (w_x,w_y) ) = ((u_x,u_y),(w_x,w_y)) + ((v_x,v_y)(w_x,w_y))
and
<br /> ( (u_x,u_y), (v_x,v_y)+ (w_x,w_y) ) = ((u_x,u_y),(v_x,v_y)) + ((u_x,u_y)(w_x,w_y))
and
(\lambda(u_x,u_y), (v_x,v_y)) = ((u_x,u_y),\lambda(v_x,v_y)) = \lambda( (u_x,u_y),(v_x,v_y))

Also, most (all?) discussions of "inner product" would include the condition $((x_1,y_1),(x_1,y_1)) \geq 0$, with $>0$ holding whenever $(x_1,y_1) \neq (0,0)$. I don't know if your textbook or notes includes this, but if it does then you need to check that as well.