Mark44 said:
A vector space generally has an infinite number of elements.
toforfiltum said:
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.
toforfiltum said:
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.
Mark44 said:
The vector space of 2 x 2 matrices has dimension 4.
toforfiltum said:
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 3
i + 5
j - 6
k or as <3, 5, -6>, omitting the unit vectors i, j, and k. I prefer the latter form, as it's easier to write.
toforfiltum said:
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.
toforfiltum said:
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.