# Linear Algebra: intersection of subspaces

• TheTangent
In summary, the problem involves finding the dimension of the intersection of two subspaces in a vector space. The dimension is affected by the bases chosen for the subspaces, as well as the dimensions of the subspaces themselves. The solution involves finding a basis for the intersection and expanding it to bases for both subspaces.
TheTangent

## Homework Statement

I'm working on a problem that involves looking at the dimension of the intersection of two subspaces of a vector space.

## Homework Equations

$M \subset V$
$N \subset V$
dim(M $\cap$ N)
$[\vec{v}]_{B_M}$ is the coordinate representation of a vector v with respect to the basis for M

## The Attempt at a Solution

I reformulated M $\cap$ N in a bunch of different ways that would be too long to copy down here, but I finally came to this (which may or may not be useful to me in my larger problem but I'm wondering if it is valid itself):

$\vec{v}$ is itself, so it must have the same dimension in both M and N, and since the bases are ordered, for each $\vec{b}_{Mi}$ in $B_M$ for which the corresponding scalar is not zero in the linear combination of elements of $B_M$ equal to $\vec{v}$, and each $\vec{b}_{Nj}$ in $B_N$ for which the corresponding scalar is not zero in the linear combination of elements of $B_N$ equal to $\vec{v}$, if i=j then $\vec{b}_{Mi}$ and $\vec{b}_{Nj}$ are dependent

and $[\vec{v}]_{B_M}$ has zeros in the same places as $[\vec{v}]_{B_N}$

but there is a major problem here with the fact that we may have dimM ≠ dimN

first imaging a basis for the intersection {b1,b2,...,bp}, then expand it to a basis of M, {b1,b2,...,bp,m1,m2,...,mq},
also expand it to a basis for N, {b1,...,bp,n1,n2,...,nr}, dimension of the intersection is p, dim(M)=p+q, dim(N)=p+r.

beautiful

## 1. What is the definition of "intersection of subspaces" in linear algebra?

The intersection of subspaces in linear algebra refers to the set of all vectors that are common to two or more subspaces. It is denoted by ∩ and can also be thought of as the overlap or shared space between different subspaces.

## 2. How is the intersection of subspaces calculated?

The intersection of subspaces is calculated by finding the common solutions to the systems of equations that define each subspace. This can be done by setting the equations equal to each other and solving for the variables. The resulting solution will be the set of vectors that make up the intersection of the subspaces.

## 3. What is the significance of the intersection of subspaces in linear algebra?

The intersection of subspaces is important in linear algebra because it allows us to understand the relationships between different subspaces. It can help us determine if two subspaces are parallel, if one subspace is contained within another, or if the subspaces are orthogonal to each other.

## 4. Can the intersection of subspaces be empty?

Yes, it is possible for the intersection of subspaces to be empty. This would occur if the subspaces do not have any common solutions or if they are parallel to each other. In this case, the intersection would be the zero vector, which represents an empty set.

## 5. How is the dimension of the intersection of subspaces related to the dimensions of the individual subspaces?

The dimension of the intersection of subspaces is always less than or equal to the dimensions of the individual subspaces. If the dimensions of the individual subspaces are equal, then the dimension of the intersection will also be equal. However, if the dimensions are not equal, then the dimension of the intersection will be the smaller of the two dimensions.

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