What Does Dimension Mean in Vector Spaces?

[toforfiltum]

I'm confused about this. I know that if the dimension of the vector space is say, 2, then there will be 2 elements, right? eg. $\left( \begin{array}{cc} 1 & 0\\ 0 & -1 \end{array} \right)$
What I want to know is if the dimension of vector space is still two if the matrix is like this:
$\left( \begin{array}{cc} 1 & 0\\0 & -1\\ 5 & 4 \end{array} \right)$

The dimensions depend on the elements, and it is shown by columns, not rows?
Thanks!

 

[Mark44]

I'm confused about this. I know that if the dimension of the vector space is say, 2, then there will be 2 elements, right?

A vector space generally has an infinite number of elements.

Could you be confused about a set of vectors that is a basis for a vector space?

eg. $\left( \begin{array}{cc} 1 & 0\\ 0 & -1 \end{array} \right)$

What is the above supposed to represent? As you wrote it, it is a matrix, and so has very little to do with a vector space of dimension 2.

The vector space of 2 x 2 matrices has dimension 4.

What I want to know is if the dimension of vector space is still two if the matrix is like this:
$\left( \begin{array}{cc} 1 & 0\\0 & -1\\ 5 & 4 \end{array} \right)$

The dimensions depend on the elements, and it is shown by columns, not rows?
Thanks!

 

[toforfiltum]

A vector space generally has an infinite number of elements.

Yes, but the vector space consists of infinite elements of that particular form, right? And are elements of the second form of matrix included in the same vector space as that of the first?

The vector space of 2 x 2 matrices has dimension 4.

I don't understand. What does dimension really mean? If the dimension of vector space is 3, must there be 3 rows in a matrix of all the elements? Like the vectors i, j and k in 3D space. I am really confused.

And to confirm, if the dimension of a vector space is 2, then there must be only 2 elements, is it? I really don't know. Is the basis in this form:
$\left( \begin{array}{cc} 1 & 0\\ 0 & -1 \end{array} \right)$ or this form:

$\left( \begin{array}{cc} 1 & 0\\0 & -1\\ 5 & 4 \end{array} \right)$
Are both correct? If not, why? I'm very confused.

 

[WWGD]

I'm confused about this. I know that if the dimension of the vector space is say, 2, then there will be 2 elements, right? eg. $\left( \begin{array}{cc} 1 & 0\\ 0 & -1 \end{array} \right)$The dimensions depend on the elements, and it is shown by columns, not rows?
Thanks!

Which are the two elements you are referring to here? Is it the entries in the matrix? If so, which? If the dimension is 2, then a basis will have two elements. What vector space are you referring to here?

 

[WWGD]

Yes, but the vector space consists of infinite elements of that particular form, right? And are elements of the second form of matrix included in the same vector space as that of the first?I don't understand. What does dimension really mean? If the dimension of vector space is 3, must there be 3 rows in a matrix of all the elements? Like the vectors i, j and k in 3D space. I am really confused.

And to confirm, if the dimension of a vector space is 2, then there must be only 2 elements, is it? I really don't know. Is the basis in this form:
$\left( \begin{array}{cc} 1 & 0\\ 0 & -1 \end{array} \right)$ or this form:

$\left( \begin{array}{cc} 1 & 0\\0 & -1\\ 5 & 4 \end{array} \right)$
Are both correct? If not, why? I'm very confused.

Dimension of a space roughly refers to the number of parameters needed to fully describe the space. The line y=x+3 is one-dimensional because a single parameter x fully determines y ( although strictly speaking, this line is not a vector space)

 

[toforfiltum]

Which are the two elements you are referring to here? Is it the entries in the matrix? If so, which? If the dimension is 2, then a basis will have two elements. What vector space are you referring to here?

2D vector space? Can the basis that have 2 elements have 3 rows? Or must it strictly be 2 rows?

 

[Mark44]

A vector space generally has an infinite number of elements.

Yes, but the vector space consists of infinite elements of that particular form, right?

What particular form? A vector space of dimension two can have many different forms. For example, a vector space could be a subspace of a higher dimension space.

And are elements of the second form of matrix included in the same vector space as that of the first?

No, not at all, if I understand what you're trying to say (which isn't very clear). Your second matrix is 3 x 2. Its columns are vectors in $\mathbb{R}^3$, a space of dimension 3.

The vector space of 2 x 2 matrices has dimension 4.

I don't understand. What does dimension really mean? If the dimension of vector space is 3, must there be 3 rows in a matrix of all the elements? Like the vectors i, j and k in 3D space. I am really confused.

Forget the matrices, which are just clouding the issue. The dimension of a space equals the number of vectors that make up a basis for that space. If you're studying vector spaces, you must have come across the term basis. Look up its definition.

A vector in, say, $\mathbb{R}^3$ can be written in a couple of different ways -- such a 3i + 5j - 6k or as <3, 5, -6>, omitting the unit vectors i, j, and k. I prefer the latter form, as it's easier to write.

And to confirm, if the dimension of a vector space is 2, then there must be only 2 elements, is it?

Again, no. There are generally an infinite number of elements. A basis can contain only two elements though, and they have to be linearly independent, and they have to span the space. Both these terms are precisely defined. Please look them up.

I really don't know. Is the basis in this form:
$\left( \begin{array}{cc} 1 & 0\\ 0 & -1 \end{array} \right)$ or this form:

$\left( \begin{array}{cc} 1 & 0\\0 & -1\\ 5 & 4 \end{array} \right)$
Are both correct? If not, why? I'm very confused.

They should not be written as matrices.
The set of vectors $\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \end{bmatrix}\}$ make up a basis for $\mathbb{R}^2$, the plane. The dimension of this space (the plane) is two.

The set of vectors $\{ \begin{bmatrix} 1 \\ 0 \\ 5 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \\4 \end{bmatrix}\}$ make up a basis for a two-dimensional subspace of $\mathbb{R}^3$. IOW, they are a basis for a plane in three dimensional space. The dimension of this subspace is two, but the vectors are three-dimensional vectors.