Representing vectors with respect to a basis

[Mr Davis 97]

I'm a little bit confused about how coordinate systems work once we have chosen a basis for a vector space. Let's take R^2 for example. It is known that if we write a vector in R^2 numerically, it must always be with respect to some basis. So the vector [1, 2] represents the point (1, 2) in the xy-plane if we take our basis vectors to be the standard ones. However, I am a bit confused by this. If we let i and j be the basis vectors, the the coordinate vector [1, 2] is more fundamentally represented by 1i + 2j. However, what then, do i and j represent? One might say that i is [1, 0] and j is [0, 1]. However, this definition seems circular, because we are then using the standard basis to define the standard basis vectors... So how are i and j defined, without referencing the standard basis to define them? The same question goes for any other basis vectors we may find. How can we define these basis vectors without having to reference the standard basis?

 

[Simon Bridge]

If we let i and j be the basis vectors, the the coordinate vector [1, 2] is more fundamentally represented by 1i + 2j. However, what then, do i and j represent? One might say that i is [1, 0] and j is [0, 1]. However, this definition seems circular, because we are then using the standard basis to define the standard basis vectors... So how are i and j defined, without referencing the standard basis to define them?

Except i and j are defined to be the standard basis vectors for the right-handed Cartesian coordinate system in R².

The same question goes for any other basis vectors we may find. How can we define these basis vectors without having to reference the standard basis?

We do not define the vectors - we use them.
A set of vectors may be used as a basis for the vector space if it has a specific set of properties.
What you seem to be wrestling with is the representation of a vector ... if we have a non-standard basis, we tend to show people what they are by representing them in terms of another basis that is standard.
In order to talk about something we have to use a representation of that thing - doesn't matter what it is. That is what language is for.
But you do not have to represent a basis vector in terms of another basis ... you can just assert it symbolically.
So i j and k are orthonormal basis vectors for R³ ... we can choose a representation where i=(1,0,0) or maybe one where i=(1,1,1)/√3.
The representation is up for grabs.

x^3 is a basis vector for polynomials ... it could be $\hat e_3 = (0,0,0,1,...)^t$

The vectors themselves are defined by the rules of algebra, not the representations we use.

 

[Mr Davis 97]

Except i and j are defined to be the standard basis vectors for the right-handed Cartesian coordinate system in R². We do not define the vectors - we use them.
A set of vectors may be used as a basis for the vector space if it has a specific set of properties.
What you seem to be wrestling with is the representation of a vector ... if we have a non-standard basis, we tend to show people what they are by representing them in terms of another basis that is standard.
In order to talk about something we have to use a representation of that thing - doesn't matter what it is. That is what language is for.
But you do not have to represent a basis vector in terms of another basis ... you can just assert it symbolically.
So i j and k are orthonormal basis vectors for R³ ... we can choose a representation where i=(1,0,0) or maybe one where i=(1,1,1)/√3.
The representation is up for grabs.

x^3 is a basis vector for polynomials ... it could be $\hat e_3 = (0,0,0,1,...)^t$

The vectors themselves are defined by the rules of algebra, not the representations we use.

So let's take functions for example, as a vector space. If we are taking $\{ \sin x, \cos x \}$ to be the basis for our vector space, then any vector in our vector space can be represented by a linear combination of sin and cos. Thus, the coordinate representation of $1 \sin x + 2\cos x$ is $\begin{bmatrix}1 \\ 2\end{bmatrix}$. The key thing is that the functions seem to be things in themselves, vectors, and coordinates are merely a representation of their linear combination. Now take $\mathbb{R}^2$ for example. Let the basis be $\{ \hat{i}, \hat{j}\}$. These are two vectors which are orthogonal and point along the x and y-axis with unit length. Then any vector in our vector space can be represented as a linear combination of i and j. Thus, the coordinate representation of $1\hat{i} + 2\hat{j}$ is $\begin{bmatrix}1 \\ 2\end{bmatrix}$. So since the functions cos and sin were "things in and of themselves," what are $\hat{i}$ and $\hat{j}$? Are they purely geometric objects?

 

[Simon Bridge]

They are vectors - which is the same kind of thing as functions or polynomials etc.
They may represent a geometric translation if you like.
For what they are, read the definition of "vector".

Some vectors are also sort-of something else... i and j are just vectors.
Treat them as entirely abstract - like any other number. You seem to be able to use them fine, where is the confusion?

 

[Mr Davis 97]

They are vectors - which is the same kind of thing as functions or polynomials etc.
They may represent a geometric translation if you like.
For what they are, read the definition of "vector".

Some vectors are also sort-of something else... i and j are just vectors.
Treat them as entirely abstract - like any other number. You seem to be able to use them fine, where is the confusion?

I guess I just don't see what the objects in R^n really are. It's obvious that functions are functions, and then their coordinate representations are in R^n. But for R^n, it is its own coordinate representation, so I just don't really see what the objects are representing

 

[Simon Bridge]

I guess I just don't see what the objects in R^n really are.

The R^n thingies are just themselves - there is no "what they really are" - they are abstract numbers with no intrinsic meaning.

It's obvious that functions are functions, and then their coordinate representations are in R^n.

So what is a function? Really?
After all 3x^2+2x+5 can be represented as $(3,2,5)^t$ though it may make you feel better writing out: $(3,2,5)(x^2,x,1)^t$ instead.
Here the (3,2,5) plays the same role in the vector representation as the 7 steps thataway plays in " 7 things". That help?
(The nit vector plays the role of "thataway"... and it need not be a direction in space.)

You are dealing with algebra at it's most abstract ... it is what you do with it that gives it meaning.

 

[Mark44]

I guess I just don't see what the objects in R^n really are.

They are vectors

It's obvious that functions are functions, and then their coordinate representations are in R^n.

When you have a vector space, there is always some field associated with the vector space. The coordinates of a vector are made up of elements from that field. The field could be the reals ($\mathbb{R}$) or the complex numbers ($\mathbb{C}$) or some finite field such as $\mathbb{Z}_2$.
The "vectors" in a vector space could be functions such as sin(t), cos(t) of your example. In this case we call this a function space. Other examples of function spaces are $p_n$, the space of polynomials of degree less than n. A spanning set (i.e. a basis) would be {1, t, t², ..., t^{n - 1}}. A function in this space can be represented as a tuple, such as <1, 0, 3> to represent 1 + 3t². Different bases are possible, which would produce a different set of coordinates.

But for R^n, it is its own coordinate representation

No, this is the wrong way to look at it. In Rⁿ each coordinate of a vector is an element of $\mathbb{R}$, the field. In the vector space $\mathbb{C}^2$, each coordinate would be a complex number -- the implied field being $\mathbb{C}$.

, so I just don't really see what the objects are representing

 

[FactChecker]

Vectors can easily exist when no coordinate system has been defined. Suppose we have one point, O, on the ground and put two points I and J on the ground at right angles one foot from O. There are vectors O→I and O→J. They exist even if no one is around to define a coordinate system. Obviously we could call the vector O→I i and call the vector O→J j. We can also make a coordinate system so that O→I = (1,0) and O→J = (0,1), but that is not necessary. It is just as valid to use a coordinate system rotated 45° so that O→I is (√2, √2) and O→J is (√2, -√2). In that case, i = (√2, √2) and j = (√2, -√2).

There is a distinction between vectors (like O→I and O→J above) that are defined independent of choice of coordinate systems and those that are defined by the coordinate system (like the coordinate basis vector defined to be (1,0) in any coordinate system.) The first type satisfy the coordinate transformations of tensors. The second type do not. The first type are physical entities, independent of the coordinate system choice. The second type are mathematical entities, completely defined by the coordinate system choice.

 

[Stephen Tashi]

So the vector [1, 2] represents the point (1, 2) in the xy-plane

That could be debated. Some would say [1,2] is the vector "from the origin to the point (1,2)".

if we take our basis vectors to be the standard ones. However, I am a bit confused by this. If we let i and j be the basis vectors, the the coordinate vector [1, 2] is more fundamentally represented by 1i + 2j. However, what then, do i and j represent? One might say that i is [1, 0] and j is [0, 1]. However, this definition seems circular

You are correct. Such a definition would be circular. The non-circular way is to define the standard basis by words that give the vectors some names (e.g. "i", "j") and gives an order to them (e.g "i" is first and "j" is second). Then the coordinate representation of vector [a,b] is defined by the convention that it represents ai + bj.

If you think of "points" as distinct from vectors, and imagine that $\mathbb{R}^2$ has a implied x-y cartesian coordinate system then you could describe the x-unit vector "i" to be the vector from (0,0) to (1,0). That wouldn't be a circular description of the vector "i".

The same question goes for any other basis vectors we may find. How can we define these basis vectors without having to reference the standard basis?

There can be other vector spaces besides $\mathbb{R}^n$ or $\mathbb{C}^n$. Have you studied any of them?