# Linear Algebra - Dimension of Kernel

• steelphantom
In summary, the dimension of kernel in linear algebra refers to the number of independent vectors that make up the null space of a matrix. It is related to the rank of a matrix through the rank-nullity theorem and is important in understanding the solutions, structure, and properties of a matrix. The dimension of kernel can be calculated using the rank-nullity theorem or by finding the null space of a matrix. Some real-world applications of the dimension of kernel include image and signal processing, data compression, and solving systems of linear equations in various fields.
steelphantom

## Homework Statement

Suppose that U and V are finite-dimensional vector spaces and that S is in L(V, W), T is in L(U, V). Prove that

dim[Ker(ST)] <= dim[Ker(S)] + dim[Ker(T)]

## Homework Equations

(*) dim[Ker(S)] = dim(U) - dim[Im(T)]
(**) dim[Ker(T)] = dim(V) - dim[Im(S)]

## The Attempt at a Solution

I know that ST is in L(U, W), so dim[Ker(ST)] = dim(U) - dim[Im(ST)]. So now I need to show:

dim(U) - dim[Im(ST)] <= dim(U) + dim(V) - dim[Im(T)] - dim[Im(S)]

But that just boils down to showing that dim[Im(ST)] is greater than dim[Im(T)] + dim[Im(S)]. It seems like I didn't really get anywhere. What am I missing? Thanks for your help.

steelphantom said:

## Homework Statement

Suppose that U and V are finite-dimensional vector spaces and that S is in L(V, W), T is in L(U, V). Prove that

dim[Ker(ST)] <= dim[Ker(S)] + dim[Ker(T)]

## Homework Equations

(*) dim[Ker(S)] = dim(U) - dim[Im(T)]
(**) dim[Ker(T)] = dim(V) - dim[Im(S)]

## The Attempt at a Solution

I know that ST is in L(U, W), so dim[Ker(ST)] = dim(U) - dim[Im(ST)]. So now I need to show:

dim(U) - dim[Im(ST)] <= dim(U) + dim(V) - dim[Im(T)] - dim[Im(S)]

But that just boils down to showing that dim[Im(ST)] is greater than dim[Im(T)] + dim[Im(S)]. It seems like I didn't really get anywhere. What am I missing? Thanks for your help.

Are you and rjw5002 taking the same course? My response to his identical question is here:

We must be taking the same course. It's the second course in Linear Algebra at Penn State. I read your post where you said the following:

Suppose v is in the null space of T. Then Tv= 0 so STv= 0 and v is in the null space of ST. That shows that the null space of T is a subspace of the null space of ST. Now suppose v is in U but NOT in the null space of T. Then Tv is a non-zero vector in V and, in order that STv=0, must be in the null space of T. That is, v is in T-1(null space of V). That set cannot have dimension larger than dim(null space of S). Putting those together, the dimension of the null space of TS cannot be larger than the dim(null space of S)+ dim(null space of T).

So I understand how you show that T is a subspace of Ker(ST), but what do you mean by "in order that STv=0, must be in the null space of T. That is, v is in T-1(null space of V)."? I'm not really sure what this is saying.

Did I say that? Of course, that's a typo. If Tv is such that S(Tv)= 0, then Tv is in the null space of S. So that v itself is in T-1(null space of S).

## What is the definition of dimension of kernel in linear algebra?

The dimension of kernel in linear algebra refers to the number of independent vectors that make up the null space of a matrix. It represents the number of free variables in a linear system of equations.

## How is the dimension of kernel related to the rank of a matrix?

The dimension of kernel and the rank of a matrix are related through the rank-nullity theorem, which states that the rank of a matrix plus the dimension of its kernel equals the number of columns in the matrix. This means that the dimension of kernel is the number of linearly independent equations that are required to fully describe the null space of a matrix.

## Why is the dimension of kernel important in linear algebra?

The dimension of kernel is important because it provides valuable information about the solutions of a linear system of equations. It allows us to determine the number of free variables and the non-trivial solutions that exist for a given set of equations. It also helps us to understand the structure and properties of a matrix.

## How can the dimension of kernel be calculated?

The dimension of kernel can be calculated using the rank-nullity theorem or by finding the null space of a matrix. The null space can be found by solving the homogeneous system of equations AX = 0, where A is the coefficient matrix and X is the vector of unknowns. The dimension of kernel is equal to the number of free variables in the solution.

## What are some real-world applications of the dimension of kernel?

The dimension of kernel has many applications in fields such as engineering, computer science, and physics. It is used in image and signal processing, data compression, and pattern recognition. It also plays a crucial role in solving systems of linear equations in real-world problems, such as optimizing resource allocation and analyzing economic data.

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