Question

Let \(T: \mathbf{M}_{n n} \rightarrow \mathbb{R}\) denote the trace map: \(T(A)=\operatorname{tr} A\) for all \(A\) in \(\mathbf{M}_{n n}\). Show that \(\operatorname{dim}(\operatorname{ker} T)=n^{2}-1\)

Short answer

The dimension of \(\operatorname{ker}(T)\) is \(n^2 - 1\).

Step-by-step solution

Understanding the Trace Map

The trace of a square matrix \(A\) is defined as \(\operatorname{tr}(A) = \sum_{i=1}^{n} a_{ii}\), which is the sum of the diagonal elements of \(A\). The trace map \(T\) takes a matrix \(A\) from the space \(\mathbf{M}_{nn}\) of \(n\times n\) matrices and gives us a real number, which is the trace of \(A\).

Identifying the Kernel of the Trace Map

The kernel of the trace map \(T\), denoted \(\operatorname{ker}(T)\), consists of all matrices \(A\) such that \(T(A) = 0\). That is, \(A\) is in \(\operatorname{ker}(T)\) if and only if \(\operatorname{tr}(A) = 0\). These are matrices whose diagonal elements sum to zero.

Determining the Dimension of \(\mathbf{M}_{nn}\)

The vector space \(\mathbf{M}_{nn}\) consists of all \(n \times n\) matrices with real entries. The dimension of this space is \(n^2\) since an \(n \times n\) matrix has \(n^2\) entries, each of which can be independently chosen.

Finding the Dimension of \(\operatorname{ker}(T)\)

We need to find the dimension of the kernel of the trace map. Using the dimension formula for linear maps, which states \(\dim(\text{domain}) = \dim(\operatorname{ker}(T)) + \dim(\operatorname{im}(T))\), we first recognize that the image of \(T\) is \(\mathbb{R}\), a 1-dimensional space. So, \(\dim(\operatorname{im}(T)) = 1\).

Applying the Dimension Formula

Substitute in the dimension values: \(n^2 = \dim(\operatorname{ker}(T)) + 1\). Solving for \(\dim(\operatorname{ker}(T))\), we get \(\dim(\operatorname{ker}(T)) = n^2 - 1\).

Key concepts

Trace of a Matrix

The trace of a matrix is a basic yet essential concept in linear algebra. It applies specifically to square matrices, which have the same number of rows and columns. The trace of a square matrix is the sum of its diagonal elements. Mathematically, if we have a square matrix \( A \) with diagonal elements \( a_{11}, a_{22}, \ldots, a_{nn} \), then the trace \( \operatorname{tr}(A) \) is calculated as \( \operatorname{tr}(A) = \sum_{i=1}^{n} a_{ii} \).

Calculating the trace is straightforward: simply add up all the diagonal elements of the matrix. For example, for a 3x3 matrix:

• A = \( \begin{pmatrix} 2 & 5 & 1 \ 0 & 3 & 4 \ 1 & 0 & 9 \end{pmatrix} \)
  
  
• The diagonal elements are 2, 3, and 9.
  
• The trace is \( 2 + 3 + 9 = 14 \).

Understanding the trace helps in many applications, such as determining the eigenvalues of a matrix, where the sum of eigenvalues equals the trace.

Kernel of a Linear Map

The kernel of a linear map is a critical concept that describes all the elements that are mapped to the zero vector in the codomain. For the specific case of the trace map \( T : \mathbf{M}_{n n} \rightarrow \mathbb{R} \), the kernel consists of all matrices \( A \) such that the trace of \( A \) is zero, i.e., \( \operatorname{tr}(A) = 0 \).

The matrices in the kernel have diagonal elements that sum to zero. Such property often makes them useful, for example, in constructing matrices that represent traceless transformations, like rotations. Visualize the kernel of a linear map as a subspace within the larger space it belongs to.

For instance, consider:

• A 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is in the kernel if \( a + d = 0 \).
• This means if you fix one diagonal element, the other must be its negative to maintain the condition.

Exploring the kernel unveils a rich structure of the linear map that can be leveraged in solving equations and analyzing transformations.

Dimension Formula for Linear Maps

The dimension formula for linear maps is an invaluable tool for understanding the relationship between a map's domain, kernel, and image. It states that for a linear map \( T: V \rightarrow W \), the dimension of the domain \( V \) is equal to the sum of the dimensions of the kernel and the image. Formally, \( \dim(V) = \dim(\operatorname{ker}(T)) + \dim(\operatorname{im}(T)) \).

This formula elegantly encapsulates the idea of how a linear map distributes the dimensions of its input space between the unaffected (kernel) and its transformed part (image).

In our context:

• The dimension of \( \mathbf{M}_{nn} \) is \( n^2 \), since we can independently assign any real number to any matrix element.
• The trace map \( T \) has an image (the real numbers \( \mathbb{R} \)), whose dimension is 1.
• By applying the dimension formula \( n^2 = \dim(\operatorname{ker}(T)) + 1 \), we find \( \dim(\operatorname{ker}(T)) = n^2 - 1 \).

This explanation underscores how dimension counting helps unravel properties of linear transformations, greatly aiding comprehension in linear algebra.