Inner Product Spaces: Testing on C3

[Hallingrad]

Hey guys,

In one of the questions for our assignment we have to decide whether <v,w> = v$$^{}T$$Aw (with a conjugate bar over w) defines an inner product on C3. We are given three 3x3 marices to test this. What is the procedure for doing this? Do we just give w and v values such as a1, a2, a3... an, or do we give them real values? If the former, how do we take into account that we are using the conjugate on w's terms?

Also, how does one find a basis for Wperp given that W is a subspace of V spanned by {1, cosx, cos2x,... cosnx}? Do we do gram schmidt on a few of the terms, or just cosnx? Thanks.

 

[lanedance]

Hey guys,

In one of the questions for our assignment we have to decide whether <v,w> = v$$^{}T$$Aw (with a conjugate bar over w) defines an inner product on C3. We are given three 3x3 marices to test this. What is the procedure for doing this? Do we just give w and v values such as a1, a2, a3... an, or do we give them real values? If the former, how do we take into account that we are using the conjugate on w's terms?

start with the defintion of an inner product on a complex space, what must it sastify?

the requirement for the conjugate should become apparent

Also, how does one find a basis for Wperp given that W is a subspace of V spanned by {1, cosx, cos2x,... cosnx}? Do we do gram schmidt on a few of the terms, or just cosnx? Thanks.

the components of Wperp are, by definition, orthogonal to every element of W, so orthogonal to every basis vector of W

 

[Hallingrad]

start with the defintion of an inner product on a complex space, what must it sastify?

the requirement for the conjugate should become apparent

the components of Wperp are, by definition, orthogonal to every element of W, so orthogonal to every basis vector of W

Yeah I know the properties of an inner product. I just don't know what procedure I should use when actually given a matrix with values that multiply two vectors v and w that have unknown values.

But how can I use gram-shmidt on an indeterminate number of vectors (i.e. 1 to n)?

 

[lanedance]

Yeah I know the properties of an inner product. I just don't know what procedure I should use when actually given a matrix with values that multiply two vectors v and w that have unknown values.

as its a property of the inner product it needs to be true for every v,w

don't assume anything for the vectors, start with applying it for arbitrary v & w then try and re-arrange things to show RHS = LHS for any v or w

pick a property and i'll help if you want

But how can I use gram-shmidt on an indeterminate number of vectors (i.e. 1 to n)?

whats is V in this question?
gram-schimdt is a process for finding an orthogonal set of vectors from a set of linearly independent vectors - are you given a set of linearly independent vectors spanning V?