# Can Linear Operators A and B Affect the Rank of AB in V?

• Zorba
In summary, the conversation discusses solving a problem involving linear operators A and B on a vector space V. The solution involves proving that the rank of the product AB is less than or equal to the rank of A. This is done by showing that the kernel of A is a subset of the kernel of AB, using the rank-nullity theorem. However, there is some uncertainty about whether the step of concluding that the kernel of A is a subset of the kernel of AB is valid.
Zorba
Studying old exam papers from my college I came across the following:

Given linear operators $$A,\,B: V\rightarrow V$$, show that:

$$\textrm{rk}AB\le \textrm{rk}A$$

My solution:
Since all $$v \in \textrm{Ker}B$$ are also in $$\textrm{Ker}AB$$ (viz $$ABv=A(Bv)=A(0)=0$$) and potentially there are $$w \in \textrm{Ker}A$$ such that $$w \in \textrm{Im}B$$ which implies $$w$$ must be also in $$\textrm{Ker}AB$$ (viz $$ABv=A(Bv)=A(w)=0$$ - either $$Bv$$ is inconsistant then we still have $$Bv=0$$, or we have $$Bv=w$$ and follows as earlier) thus we have $$\textrm{Ker}A \subset \textrm{Ker}AB$$ and thus from the rank-nullity theorem it follows that $$\textrm{rk}A \ge \textrm{rk}AB$$.

Is everything I've done there legal? I'm a bit iffy about the step where I conclude $$\textrm{Ker}A \subset \textrm{Ker}AB$$...

Having w in Ker A and w in I am B does not imply that w is in Ker AB (although if w = Bv then v is in Ker AB, since ABv = Aw = 0).

Hint: You can solve this by considering only images, not kernels.

## 1. What is a linear operator?

A linear operator is a mathematical function that maps a vector space to itself in a linear fashion. It takes in a vector and outputs another vector, preserving the basic properties of addition and scalar multiplication.

## 2. What is the role of linear operators in linear algebra?

Linear operators play a crucial role in linear algebra as they allow us to perform operations on vectors and matrices in a more efficient and structured manner. They also help us to understand the properties and behavior of vector spaces.

## 3. How are linear operators and matrices related?

Linear operators and matrices are closely related as every linear operator can be represented by a matrix and vice versa. The matrix representation of a linear operator depends on the basis chosen for the vector space.

## 4. What is the rank of a linear operator?

The rank of a linear operator is the dimension of its range, which is the set of all possible outputs of the operator. It can also be defined as the number of linearly independent columns or rows in the matrix representation of the operator.

## 5. How does the rank of the product of two linear operators relate to the ranks of the individual operators?

The rank of the product of two linear operators, A and B, is less than or equal to the minimum of the ranks of the individual operators. In other words, rk(AB) ≤ min(rk(A), rk(B)). This result is known as the rank inequality theorem.

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