No, I didn't say that at all. What does any two-dimensional space look like geometrically?helix999 said:you mean to say that V and W are 2x2 matrices and their intersection would also be a 2x2 matrix...so the required dimension is 2 only?
Mark44 said:What does any two-dimensional space look like geometrically?
helix999 said:I am sorry but i am not able to grasp the idea u r trying to put.
Mark44 said:Considering the real plane, what is the dimension of a single point? What is the dimension of a line?
Mark44 said:You don't need to be concerned about the fact that V and W are subspaces, nor that they are embedded in a four-dimensional space.
V and W are two-dimensional, so you know what they look like geometrically, don't you? Now, what are all of the possibilities for the intersection of V and W?
helix999 said:it is single dimension
Dick said:You do need to pay some attention to the dimension of the space they are embedded in. If they were embedded in three space a 0 dimensional intersection would be impossible.
Mark44 said:I asked two questions. Your answer above is the answer to which question?
Mark44 said:You don't need to be concerned about the fact that V and W are subspaces, nor that they are embedded in a four-dimensional space.
Dick said:You do need to pay some attention to the dimension of the space they are embedded in. If they were embedded in three space a 0 dimensional intersection would be impossible.
why is it impossible for subspaces?HallsofIvy said:You also need to pay attention to the fact that they are subspaces. The planes (x, y, 0, 0) and (x, y, 0 , 1) have empty intersection. That is impossible for subspaces.
Dick said:If they were embedded in three space a 0 dimensional intersection would be impossible.
helix999 said:How come a point can exist as an intersection of subspaces in 4 space but not in 3 space?
In other words, "what is the dimension of the kernel of T" or "what is the nullity of T".helix999 said:those were the new concepts for me...thnx
How do I proceed for such type of questions:
Let V be the real vector space of all real 2 x 3 matrices, and let W be the real vector space of all real 4 x 1
column vectors. If T is a linear transformation from V onto W, what is the dimension of the subspace
{v belongs to V:T(v)=0}?
helix999 said:I got the answer of my question, really appreciate ur posts for making it clear to me. Thnx a bunch!
why is it impossible for subspaces?
No, we do NOT substitute x^2 as tan theta. If we did then we would have 2 xdx= sec2 theta dtheta- and there is no "x" outside the square root to use with dx.helix999 said:That was the complete question and i guess i understood the explanation.
We solve the integrals like sqrt(1+x^4) dx (limit 0 to 1) by substituting x^2 as tan theta but how do we proceed for integrals like sqrt(1-x^4) dx (limit 0 to 1)?
HallsofIvy said:In other words, "what is the dimension of the kernel of T" or "what is the nullity of T".
Is that really the entire question? There are many possible answers. For example, T(v)= 0 for all v is a linear transformation and it's kernel is all of V. Since V is a 6 dimensional space (Taking Vij to be the matrix with 1 at "ith row, jth column" and 0 everywhere else, for i= 1 to 2, j= 1 to 3 gives a basis), so that in this example, the dimension of the kernel would be 6.
On the other hand the linear transformation that maps
[tex]\left[\begin{array}{ccc} a & b & c \\d & e & f\end{array}\right]\right)[/tex] to [tex]\left[\begin{array}{c} a \\ 0 \\ 0 \\ 0\end{array}\right][/tex]
has nullity 5: dimension of the kernel is 5.
Perhaps the problem is "what is the smallest possible dimension of the kernel of T".
Since the largest the image of T can be is 4 dimensional, the dimension of "the real vector space of all real 4 x 1 column vectors", the smallest the kernel can be is 6- 4= 2.
Subspaces are mathematical objects that are a subset of a larger vector space. They have the same properties as the larger vector space and can be thought of as smaller versions of the larger space.
The dimension of a subspace is the number of linearly independent vectors that span the subspace. It represents the number of directions or degrees of freedom within the subspace.
The dimension of a subspace can be found by determining the number of linearly independent vectors in the subspace. This can be done by using techniques such as Gaussian elimination or finding the null space of a matrix.
Yes, a subspace can have a dimension of zero. This means that the subspace only contains the zero vector and has no other linearly independent vectors. In this case, the subspace is referred to as the trivial subspace.
The dimensions of subspaces are closely related to linear transformations. The dimension of the image of a subspace under a linear transformation will be equal to or less than the dimension of the subspace itself. Additionally, the dimension of the null space of a linear transformation will be equal to the difference between the dimension of the domain and the dimension of the image.