91Ó°ÊÓ

Vector spaces are fundamental structures in linear algebra. They consist of a set of vectors, along with operations of vector addition and scalar multiplication that satisfy certain axioms.
Vector spaces can be finite or infinite dimensional, depending on the number of vectors in the basis. A basis of a vector space is a set of vectors that is linearly independent and spans the entire space.
In the problem, we have two finite-dimensional vector spaces, \(\text{dim}(L_1) = n\) and \(\text{dim}(L_2) = m\). We choose bases \(B_1 = \{e_1, e_2, \ldots, e_n\}\) for \(\bar{L}_1\) and \(B_2 = \{f_1, f_2, \ldots, f_m\}\) for \(\bar{L}_2\).
These bases help transform vector operations into matrix operations, which simplifies calculations and proofs significantly.

Linear transformations are mappings between vector spaces that preserve the operations of vector addition and scalar multiplication. Mathematically, a linear transformation \(\text{\textphi}: L_1 \rightarrow L_2\) adheres to:

• \( \text{\textphi}(x + y) = \text{\textphi}(x) + \text{\textphi}(y) \)
• \( \text{\textphi}(c x) = c \text{\textphi}(x) \)

where \( x \) and \( y \) are vectors in \( L_1 \), and \( c \) is a scalar.
In our problem, \( \text{\textphi} \) and \( \text{\textpsi} \) are linear transformations between \( L_1 \) and \( L_2 \). These transformations can be represented by matrices once we fix bases in \( L_1 \) and \( L_2 \). The use of matrix representation makes it easier to manipulate and study these transformations.

Matrix representation of linear transformations facilitates their manipulation and provides a concrete form. For any linear transformation \( \text{\textphi}: L_1 \rightarrow L_2 \), the matrix representation \( A \) is formed by expressing the images of the basis vectors in \( L_1 \) as linear combinations of the basis vectors in \( L_2 \).
Let \( \text{\textphi}(e_i) = \sum_{j}a_{ji}f_j \). Then, \( A = (a_{ji}) \), where each entry corresponds to a coordinate that describes how \( \text{\textphi} \) transforms the basis vector from \(\bar{L}_1\) to \(\bar{L}_2\).
In our problem, \( \text{\textphi} \) is represented by a matrix \( A \) of size \( m \times n \) and \( \text{\textpsi} \) by a matrix \( B \) of size \( n \times m \). The compositions \( \text{\textphi}\text{\textpsi} \) and \( \text{\textpsi}\text{\textphi} \) correspond to the products \( AB \) and \( BA \), respectively.

The trace of a square matrix is the sum of the elements on its main diagonal. One of the fascinating properties of the trace function is its cyclic property: \( \text{Tr}(XY) = \text{Tr}(YX) \) for matrices \( X \) and \( Y \) of appropriate dimensions.
This property is pivotal in many linear algebra proofs and simplifies the analysis of matrix products.
In our context, it shows immediately that \( \text{Tr}(\text{\textphi}\text{\textpsi}) = \text{Tr}(AB) \), which equals \( \text{Tr}(BA) = \text{Tr}(\text{\textpsi}\text{\textphi}) \). Thus, the traces of the compositions of our transformations in different orders are equal, demonstrating the required result.

In mathematics, proof techniques are methods or strategies used to establish the truth of a statement. For linear algebra problems, common techniques include:

• Direct proofs, where the statement is demonstrated using logical deductions from axioms and previously established results.
• Matrix manipulation, such as using properties of matrix operations.
• Using specific properties of mathematical objects, like the cyclic property of the trace.

In our proof, we utilized matrix representations and the cyclic property of the trace to show that \( \text{Tr}(\text{\textphi}\text{\textpsi}) = \text{Tr}(\text{\textpsi}\text{\textphi}) \). This approach showcases how exploring intrinsic properties of matrices can lead to elegant and simplified solutions to seemingly complex problems.