Question

Show that the intersection of two invariant subspaces of an operator is also an invariant subspace.

Short answer

The intersection of two invariant subspaces of a linear operator is also an invariant subspace.

Step-by-step solution

Define the Invariant Subspaces

Let $V$ be a vector space and $T: V \rightarrow V$ a linear operator. Assume $U_1$ and $U_2$ are both invariant subspaces of $V$ under $T$, i.e., $T(U_1) \subseteq U_1$ and $T(U_2) \subseteq U_2$.

Denotes the Intersection

Let $U = U_1 \cap U_2$ denote the intersection of the two invariant subspaces $U_1$ and $U_2$. The challenge is to show that $T(U) \subseteq U$, which signifies $U$ being an invariant subspace.

Select an Arbitrary Vector

Select an arbitrary vector $v$ from $U$. Thus, $v$ belongs to both $U_1$ and $U_2$ since it's part of their intersection.

Apply the Operator

By our definition of an invariant subspace, $T(v)$ has to belong to both $U_1$ and $U_2$ because $v$ belongs to both of these subspaces. Therefore, $T(v)$ is in the intersection $U$.

Conclude the Proof

Since this result holds for any arbitrary vector in the intersection $U$, it means $T(U) \subseteq U$, therefore the intersection $U$ is also an invariant subspace.

Key concepts

Linear Operator

A linear operator is a key concept in the study of linear algebra and vector spaces. It's a rule, or a function, denoted by T, that takes vectors from a vector space and maps them right back to that same space. More formally, for any vector space V, a linear operator T is a function \(T: V \rightarrow V\) with two special properties: additivity and homogeneity of degree 1.

Let us take two vectors \(x\) and \(y\) in V, and a scalar \(\alpha\), the following conditions must be true for T to be a linear operator:

• Additivity: \(T(x+y) = T(x) + T(y)\)
• Homogeneity: \(T(\alpha x) = \alpha T(x)\)

These properties ensure that the structure of the vector space is preserved after the application of T. This is crucial for the concept of invariant subspaces, which are essentially chunks of the vector space that are 'compatible' with the operator in the sense that applying the operator to any vector in the subspace yields another vector still within the subspace.

Understanding linear operators is also fundamental when dealing with matrices, differential equations, and transformations, all of which are frequent topics in more advanced studies in mathematics and engineering.

Vector Space

At its core, a vector space is a collection of objects known as vectors, which can be added together and multiplied by scalars (numbers), subject to certain axioms. The beauty of vector spaces lies in their generality: they can consist of traditional 2D or 3D vectors, functions, polynomials, or even matrices.

There are several axioms that a vector space must satisfy, which define its structure and the nature of vector addition and scalar multiplication:

• Associativity of addition
• Commutativity of addition
• Existence of an additive identity
• Existence of additive inverses
• Distributivity of scalar multiplication with respect to vector addition
• Distributivity of scalar multiplication with respect to field addition
• Compatibility of scalar multiplication with field multiplication
• Existence of a multiplicative identity

When a vector space is equipped with a linear operator, new dynamics come into play. Subspaces start to become relevant, especially the ones stable under the operation—these are the invariant subspaces we're discussing.

Subspace Intersection

Subspace intersection is a nuanced concept that combines the notions of vector spaces and subspaces. A subspace is a smaller 'space' inside a vector space that is also a vector space itself, obeying all the axioms mentioned earlier. Now, when we speak of an intersection, \(U = U_1 \cap U_2\), we are referring to the set of all vectors that \(U_1\) and \(U_2\) have in common.

In the case of invariant subspaces, the intersection retains an interesting property. If \(T\) is a linear operator for which \(U_1\) and \(U_2\) are invariant, then any vector in their intersection remains in the intersection even after \(T\) is applied. This sturdiness of the intersection under the linear operator is what makes the intersection itself an invariant subspace.

This concept has important implications in various fields like differential equations, where solving a system sometimes involves finding stable solutions that correspond to invariant subspaces, and in quantum mechanics, where invariant subspaces can represent states that are stable under certain operations. The intersection principle ensures that you can work within smaller, more manageable chunks of a vector space while keeping the properties that are relevant to the structure and behavior of the system you're studying.