Intersection of subspaces

In summary, the subspaces U and V of R³ intersect at U = V, with the intersection being 1-dimensional. The original restrictions still apply to the intersection, and a non-zero point that satisfies both sets can be found.
  • #1
loli12
I have 2 subspaces U and V of R^3 which
U = {(a1, a2, a3) in R^3: a1 = 3(a2) and a3 = -a2}
V = {(a1, a2, a3) in R^3: a1 - 4(a2) - a3 = 0}

I used the information in U and substituted it into the equation in V and I got 0 = 0. So, does it mean that the intersection of U and V is the whole R^3 which has no restrictions on a1, a2 and a3 (they are free)? Or do the original restrictions on both the original subspaces still being applied to the intersection?
 
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  • #2
The intersection of U and V cannot possibly be all of R³. How could the intersection of two sets be bigger than both of the sets? Both subspaces are 1-dimensional, so the intersection is either 1-dimensional or 0-dimensional. Can you find a non-zero point that is in both U and V? If so, then the intersection of U and V is U and is also V (i.e. U = V). A point in U takes the form (x, x/3, -x/3). Would such a point be in V?

x - 4(x/3) - (-x/3) = x - (4/3)x + (1/3)x = 0

so the answer is "yes."
 
  • #3


The intersection of two subspaces, U and V, is defined as the set of all vectors that are in both U and V. In this case, the intersection of U and V would be the set of all vectors that satisfy the conditions of both U and V.

Using the given information, we can see that the conditions for U and V are not contradictory, as substituting the conditions of U into the equation of V results in 0=0. This means that all vectors in U are also in V, and vice versa. Therefore, the intersection of U and V is the set of all vectors in R^3 that satisfy the conditions of both U and V.

In terms of the restrictions, the original restrictions for both U and V are still being applied to the intersection. This means that any vector in the intersection must satisfy the conditions of both U and V, which are the restrictions given in the problem. So, the intersection is not the whole R^3, but rather a subset of R^3 that satisfies both sets of restrictions.
 

What is the definition of "intersection of subspaces"?

The intersection of subspaces refers to the set of all elements that are common to two or more subspaces. In other words, it is the collection of all vectors that satisfy the defining conditions of each subspace.

How is the intersection of subspaces related to linear algebra?

The concept of intersection of subspaces is fundamental to linear algebra as it allows us to study the properties and relationships between different subspaces. It also plays a crucial role in solving systems of linear equations and understanding the geometric interpretation of linear transformations.

What is the dimension of the intersection of two subspaces?

The dimension of the intersection of two subspaces can be at most the minimum of the dimensions of the two subspaces. In other words, if one subspace has a dimension of m and the other has a dimension of n, then the dimension of their intersection can range from 0 to min(m, n).

How can we determine if two subspaces intersect at a single point or not?

If the dimension of the intersection of two subspaces is 0, then they do not intersect at any point. However, if the dimension is greater than 0, then the subspaces intersect at least at one point. To determine if they intersect at a single point, we need to check if the dimension of the intersection is equal to the sum of the dimensions of the two subspaces minus their common dimension.

What is the significance of the intersection of subspaces in real-world applications?

The concept of intersection of subspaces has various applications in fields such as computer graphics, image processing, and data analysis. For example, in image processing, the intersection of subspaces can help in identifying the common features between different images, which can then be used for image recognition and pattern recognition. It also plays a crucial role in dimensionality reduction techniques used in data analysis and machine learning.

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