In #1 of the first proof, there's a zero vector with (x) immediately after. I'm confused as to what this means. Does it mean 0 times all the x's in the function (and this way the function equals 0)? Furthermore, what does the 0 function mean?
+mech_builder Instead of f, I just called it 0_hat. The 0 indicates that it is the zero function, the hat indicates that it is a vector also. You could have called it anything, say h. The zero function takes every x and sends it to zero, so h(x) = 0 for ALL x, that's all it is:)
In the case of W={(a,b,c): a is greater than or equal to zero}. if the vector is scaled by negative scalar then W won't be subspace because its components should be greater than or equal to zero.In your case, If I am scaling the function by -1 then cf(-x)= -1 - f(x)= f(x) which does not satisfy the condition for odd function. What am I missing here.
you proved all odd functions form subspace , but what about sinx which is an odd function sin(x1)+sin(x2)=!! sin(x1+x2) clearly the condition CA is not satisfied ,? what is the mistake here?
yes, all odd functions form subspace, f(x) = sin x which is an odd function and g(x) = x which is an odd function, so (f+g)(x) = - (f+g)(-x). You get the point ? It's about the function, not the x variable.
thank you,you made it clear :) now i know how to do the same thing for even functions
I did that now. It is also a subspace and according to theorem vector space of its own. can you check and confirm
How do you show that the sum of subspaces of odd and even functions gives a vector space?
Although the zero vector function in the beginning is indeed an odd function, I think your proof showed that it is an even function.
Yes.. Even I think so!
actually zero is everything in that particular time Mr.
O(-x) = 0(x) proves that its an even function ... ? To complete the proof you have to establish tge fact that -0(x) = 0(x).
In #1 of the first proof, there's a zero vector with (x) immediately after. I'm confused as to what this means. Does it mean 0 times all the x's in the function (and this way the function equals 0)? Furthermore, what does the 0 function mean?
+mech_builder Instead of f, I just called it 0_hat. The 0 indicates that it is the zero function, the hat indicates that it is a vector also. You could have called it anything, say h. The zero function takes every x and sends it to zero, so h(x) = 0 for ALL x, that's all it is:)
If you made 1000 examples I will be happy to watch.....
:)
at around 1:00
should x and y also be elements of V?
isn't the definition you gave for a subspace the same as the definition of a linear space?
Thanks, pretty helpful
awesome!!
could you also say: "zero function": R-->{0} ?
In the case of W={(a,b,c): a is greater than or equal to zero}. if the vector is scaled by negative scalar then W won't be subspace because its components should be greater than or equal to zero.In your case, If I am scaling the function by -1 then cf(-x)= -1 - f(x)= f(x) which does not satisfy the condition for odd function. What am I missing here.
you proved all odd functions form subspace , but what about sinx which is an odd function
sin(x1)+sin(x2)=!! sin(x1+x2)
clearly the condition CA is not satisfied ,?
what is the mistake here?
yes, all odd functions form subspace, f(x) = sin x which is an odd function and g(x) = x which is an odd function, so (f+g)(x) = - (f+g)(-x). You get the point ? It's about the function, not the x variable.