We didn't touch the <u+v, w> = <u, w> + <v,w> property in class but it totally makes sense geometrically. Looking at it from how we talked about thinking of inner product like casting a shadow of one vector over the other, the addition of two vectors would cast a shadow whose length would be the summation of the individual shadows cast by u and v. I wish there was a way to upload an image!
meranara
I agree with the comment above about the shadows cast individually by u and v summed up would/should be equal to the shadow cast by the vector u+v. But, for some reason, this property seems a little off to me when just looking at it. What would be the case if the expression is <u+v,w+x>? Would it simplify to <u,w+x> + <v,w+x> = <u,w>+<u,x> + <v,w>+<v,x>?
Parker
I think that would be the case, meranara. If we substitute 'w+x' for a single variable representing that value, like 'z', then it becomes similar to the slide example to see how <u+v, z> becomes <u,z> + <v,z> which is equivalent to <u,w+x> + <v,w+x>. Since we know that the order of the vectors in an inner product don't matter, the steps from there are roughly the same as the ones in the slides.
We didn't touch the <u+v, w> = <u, w> + <v,w> property in class but it totally makes sense geometrically. Looking at it from how we talked about thinking of inner product like casting a shadow of one vector over the other, the addition of two vectors would cast a shadow whose length would be the summation of the individual shadows cast by u and v. I wish there was a way to upload an image!
I agree with the comment above about the shadows cast individually by u and v summed up would/should be equal to the shadow cast by the vector u+v. But, for some reason, this property seems a little off to me when just looking at it. What would be the case if the expression is <u+v,w+x>? Would it simplify to <u,w+x> + <v,w+x> = <u,w>+<u,x> + <v,w>+<v,x>?
I think that would be the case, meranara. If we substitute 'w+x' for a single variable representing that value, like 'z', then it becomes similar to the slide example to see how <u+v, z> becomes <u,z> + <v,z> which is equivalent to <u,w+x> + <v,w+x>. Since we know that the order of the vectors in an inner product don't matter, the steps from there are roughly the same as the ones in the slides.