Components of a vectors with respect to basis are unique?

[Ashley1nOnly]

Mentor note: Moved from Intro Physics, as this is more of a mathematics question

Homework Statement

With respect to a prescribed basis...
|e 1> |e 2> ...|e n>
Any given vector
|a> = a1|e1> +a2|e2>+...+a n|e n>,

Is uniquely represented by the (order) n-ruble of its components.
|a> <--->( a1,a2,...a n)

Homework Equations

N/A

The Attempt at a Solution

I'm trying to figure out what this means. A basis is the set of linearly independent vectors that span a space. When it's linearly independent it cannot be written as a linear combination of vectors.

Question:
How are we writing the basis as n-type of its components when linearly independent vectors cannot be written as a linear combination of vectors?

How can we represent it as a combination of its components?
How do we prove that the components are unique with respect to a given basis?

 

[Mark44]

Homework Statement

With respect to a prescribed basis...
|e 1> |e 2> ...|e n>
Any given vector
|a> = a1|e1> +a2|e2>+...+a n|e n>,

Is uniquely represented by the (order) n-ruble of its components.

That would be n-tuple. A ruble is a unit of currency in Russia.

|a> <--->( a1,a2,...a n)

Homework Equations

N/A

The Attempt at a Solution

I'm trying to figure out what this means. A basis is the set of linearly independent vectors that span a space. When it's linearly independent it cannot be written as a linear combination of vectors.

What you wrote isn't clear. In the phrase "When it's linearly independent ..." what does "it's" refer back to? The space? The set of vectors? The basis?
In any case, whatever you meant is incorrect. If you have a basis for some (vector) space , then any vector in that space can be written as a linear combination of the vectors in the basis. What this problem is about is showing that that linear combination, i.e., a1|e1> +a2|e2>+...+a n|e n>, is unique. IOW, that no other constants a1, a2, ..., an work.

Question:
How are we writing the basis as n-type of its components when linearly independent vectors cannot be written as a linear combination of vectors?

?
Look in your book for the definitions of these terms: basis, set of linearly independent vectors, set of linearly dependent vectors, spanning set.

How can we represent it as a combination of its components?
How do we prove that the components are unique with respect to a given basis?

By assuming that a given vector can be written in two ways; i.e.,
v = a1|e1> +a2|e2>+...+an|en>
and
v = b1|e1> +b2|e2>+...+bn|en>
If you reach a contradiction, that means your assumption that the vector could be written in two ways. Since that assumption turns out to be incorrect, it has to be the case that there is only one way of writing the vector. IOW, the components are unique.

 

[Ashley1nOnly]

K-hat j-hat and I-hat are linearly independent of each other. A set of linearly independent vecators that spans a space is called a basis.

So are we saying that we can't use any combination of scalars to represent the linear independent vector?

 

[Ashley1nOnly]

Okay, I see what you are saying. I had to think a little, now it makes since as to why the book says it's easier to work with the components than with the abstract vectors themselves.

 

[Mark44]

K-hat j-hat and I-hat are linearly independent of each other. A set of linearly independent vecators that spans a space is called a basis.

Yes, more or less. Your k, j, and l vectors are linearly independent. You don't have to add "of each other." Linear independence is always about a set of vectors.

So are we saying that we can't use any combination of scalars to represent the linear independent vector?

"Linear independent vector" makes no sense. You're not representing a "linear independent" vector -- it's just a vector belonging to whatever space you're working in.

What the problem is saying is that if you have a basis (a set of linearly independent vectors that spans the space), then any vector in that space can be represented as a linear combination (i.e., a sum of scalar multiples of) the vectors in the basis.

For example, the usual basis vectors for $\mathbb{R}^3$ are $\vec e_1 = <1, 0, 0>$, $\vec e_2 = <0, 1, 0>$, and $\vec e_3 = <0, 0, 1>$. Any vector in $\mathbb R^3$ can be represented as some linear combination of these three basis vectors. The vector <3, -2, 5> can be represented as $3\vec e_1 +(-2)\vec e_2 + 5\vec e_1$. <3, -2, 5> can't be represented in any other way, in terms of this basis.

What you need to show is that for a given basis and a given vector in the space spanned by the basis, there is only one representation in terms of that basis. I provided a way to do that in my last post.