MATHEMATICS S6 UNIT 6: Intersection and Sum of Subspaces.

In linear algebra, the concepts of Intersection and Sum of Subspaces are fundamental operations that allow us to combine and relate different subspaces within a larger vector space.

Let V be a vector space over a field F, and let U and W be two subspaces of V.

Intersection of Subspaces (U∩W)

Definition: The intersection of two subspaces U and W, denoted U∩W, is the set of all vectors that are common to both U and W.

                                                           U∩W= {v∈V ∣v∈U and v∈W}

Property: U∩W is always a Subspace.

 A key property is that the intersection of any two (or more) subspaces is always a subspace itself. To prove this, we can use the subspace test:

1. Contains the zero vector: Since U and W are subspaces, they both contain the zero vector 0V​. Therefore, 0V​∈ U∩W.
2. Closed under vector addition: Let v1​, v2​∈U∩W. This means v1​∈U, v1​∈W, v2​∈U, and v2​∈W. Since U is a subspace, v1​+v2​∈U. Since W is a subspace, v1​+v2​∈W. Thus, v1​+v2​∈U∩W.
3. Closed under scalar multiplication: Let v∈U∩W and c∈F (a scalar). This means v∈U and v∈W. Since U is a subspace, cv∈U. Since W is a subspace, cv∈W. Thus, cv∈U∩W.

Since all three conditions are met, U∩W is a subspace of V.

Example: In R³, let U be the x-y plane (U= {(x, y,0) ∣x,y∈R}) and W be the y-z plane (W={(0,y,z) ∣y,z∈R}). Their intersection U∩W would be the y-axis: U∩W= {(0, y,0) ∣y∈R}. This is indeed a subspace of R³.

Sum of Subspaces (U+W)

Definition: The sum of two subspaces U and W, denoted U+W, is the set of all possible sums of a vector from U and a vector from W.

                                                             U+W= {u+w ∣u∈U and w∈W}

Property: U+W is always a Subspace The sum of two subspaces is also always a subspace. We can prove this using the subspace test:

1. Contains the zero vector: Since U and W are subspaces, 0V​∈U and 0V​∈W. Therefore, 0V​=0V​+0V​∈U+W.
2. Closed under vector addition: Let v1​, v2​∈U+W. By definition, v1​=u1​+w1​ for some u1​∈U, w1​∈W, and v2 ​=u2​+w2​ for some u2​∈U, w2​∈W. Then v1​+v2 ​=(u1​+w1​) +(u2​+w2​) = (u1​+u2​) +(w1​+w2​). Since U is a subspace, u1​+u2​∈U. Since W is a subspace, w1​+w2​∈W. Thus, (u1​+u2​) +(w1​+w2​) ∈U+W.
3. Closed under scalar multiplication: Let v∈U+W and c∈F. By definition, v= u+w for some u∈U, w∈W. Then cv=c(u+w) =cu+cw. Since U is a subspace, cu∈U. Since W is a subspace, cw∈W. Thus, cu+cw ∈U+W.

Since all three conditions are met, U+W is a subspace of V. The sum U+W is often described as the smallest subspace that contains both U and W. It’s equivalent to the span of the union of bases for U and W.

Example: In R³, let U be the x-axis (U={(x,0,0) ∣x∈R}) and W be the y-axis (W= {(0, y,0) ∣y∈R}). Their sum U+W would be the x-y plane: U+W= {(x, y,0) ∣x,y∈R}.

Dimension Formula (Grassmann’s Formula)

For finite-dimensional vector spaces, there’s an important relationship between the dimensions of U, W, U∩W, and U+W:

                                        dim(U)+dim(W)=dim(U∩W) + dim(U+W)

This formula is extremely useful for finding the dimension of the sum or intersection if the other dimensions are known.

Example using the formula: Using the previous example in R³:

• U: x-y plane, dim(U)=2
• W: y-z plane, dim(W)=2
• U∩W: y-axis, dim(U∩W) =1

Using the formula: dim(U+W) = dim(U)+dim(W)−dim(U∩W) =2+2−1=3. This matches, as U+W in this case would be all of R³, which has dimension 3.

Direct Sum (U⊕W)

A special case of the sum of subspaces is the direct sum.

Definition: The sum U+W is called a direct sum, denoted U⊕W, if and only if their intersection is the zero vector only: U∩W={0V​}

Properties of Direct Sums:

• If V=U⊕W, then every vector v∈V can be written uniquely as a sum v =u+w, where u∈U and w∈W.
• For a direct sum, the dimension formula simplifies: dim(U⊕W) = dim(U)+dim(W). This makes sense because there’s no “overlap” (beyond the zero vector) to subtract.
• The concept of direct sum is crucial for decomposing a vector space into simpler components.

Example of Direct Sum: In R³, let U be the x-axis (U={(x,0,0) ∣x∈R}) and W be the y-z plane (W= {(0, y,z) ∣y,z∈R}).

• U∩W= {(0,0,0)}, so their intersection is just the zero vector.
• Therefore, their sum is a direct sum: R³=U⊕W. Every vector (a,b,c) in R³ can be uniquely written as (a,0,0)+(0,b,c).
• dim(U)=1, dim(W)=2.
• dim(U⊕W) =1+2=3, which is dim(R³).

Understanding the intersection and sum of subspaces is crucial for comprehending the structure of vector spaces, linear transformations, and many other advanced topics in linear algebra.