Question

In: Advanced Math

V is a subspace of inner-product space R3, generated by vector u =[1 1 2]T and...

V is a subspace of inner-product space R3, generated by vector

u =[1 1 2]T and v =[ 2 2 3]T.

T is transpose

(1) Find its orthogonal complement space V┴ ;

(2) Find the dimension of space W = V+ V┴;

(3) Find the angle q between u and v; also the angle b between u and normalized x with respect to its 2-norm.

(4) Considering v’ = av, a is a scaler, show the angle q’ between u and v’

Solutions

Expert Solution

Problem (1)

Note that and are non-zero vectors and one is not a scalar multiple of another.

SInce  , so,

This is a side remark if you know that

Then,

i.e., the orthogonal complement of is one dimensional.

So, any non zero vector in is a basis of

Continuing back to our problem

Let

So, it is orthogonal to every vector in , in particular to and ,

i.e.,

Putting this in equation , we obtain .

This in fact proves that is one dimensional as we claimed above.

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Problem (2)

By definition,

Now, by dimension formula, we obtain

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Problem (3)

Let be the angle between and .

Now, we know that  

Now, the problem does not mention what is.

Now, if is mistakenly typed , then the angle does not change whether you scale it by a positive constant.

So, the answer still remains the same.

So, I am assuming that is the unit vector spanning . If there is any confusion, please mention it in the comments so that I can correct myself. Thank you. I will proceed with this.

i.e.,

Let be the angle between and .

Now, we know that  

But since and .

since are non-zero vectors

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Problem (4)

Let

If , then , but it does not make sense to talk about angle between zero vector and .

So, assume that is non-zero.

Let be the angle between and .

Now, again we have,

since are non-zero vectors

Case (1):

Then since

Case (2):

Then since

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Colby Messinger answered 2 years ago
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