We know from Lie representation theory that the Lie algebra is a vector space. Therefore a representation of the Lie group can be transformation of this vector space itself which we call the Adjoint Representation. An element of this vector space, is itself represented by a matrix. For example, in the case of $SU(3)$, to show the adjoint representation, we write $V\psi(x)V^{1}$, where $V$ is a $3*3$ special unitary matrix, $\psi(x)$ is also a $3*3$ (traceless) matrix, i.e. an element in the Lie algebra vector space. So we are multiplying three $3*3$ matrices. A new vector, or $3*3$ matrix, is produced, which belongs to or is an element in the Lie algebra vector space.

But in a SM textbook I see another definition/use of the Adjoint Representation. This time we consider the (abstract) commutation relations among the generators of the $SU(3)$ and interpret them as a vector acts on the another to produce a new vector. In other words, use the structure constants as elements to build a $8*8$ matrix to represent the Lie algebra vectors: $L_{1},...L_{8}$. Now if we consider an octet, or an 8 component column object, $\psi(x)=(\psi_{1}, ..., \psi_{8})^T$ each of $\psi_{i}$ a fermion or Dirac spinor, with four complex components, i.e. spinor indices suppressed, we can write for example $L_{1}\psi(x)$ to make or show a transformation of the fermion fields.

Now my question is that in the example of $SU(3)$ are these two different Adjoint Representations related somehow, as one of them involves the operation of $3*3$ matrices and the another one involves the operation of $8*8$ matrices?

My first impression is that in the first case, the 3 dimensional one, we are using the adjoint representation of the group using the Lie algebra as a vector space whereas in the second case, the 8 dimensional one, we are using the adjoint representation of the Lie algebra using the Lie algebra itself as a vector space. And both of them are useful in writing the Lagrangian density of the SM. Is this correct?

We know from Lie representation theory that the Lie algebra is a vector space. Therefore a representation of the Lie group can be transformation of this vector space itself which we call the Adjoint Representation. An element of this vector space, is itself represented by a matrix. For example, in the case of $SU(3)$, to show the adjoint representation, we write $V\psi(x)V^{-1}$, where $V$ is a $3*3$ special unitary matrix, $\psi(x)$ is also a $3*3$ (traceless) matrix, i.e. an element in the Lie algebra vector space. So we are multiplying three $3*3$ matrices. A new vector, or $3*3$ matrix, is produced, which belongs to or is an element in the Lie algebra vector space.

But in a SM textbook I see another definition/use of the Adjoint Representation. This time we consider the (abstract) commutation relations among the generators of the $SU(3)$ and interpret them as a vector acts on the another to produce a new vector. In other words, use the structure constants as elements to build a $8*8$ matrix to represent the Lie algebra vectors: $L_{1},...L_{8}$. Now if we consider an octet, or an 8 component column object, $\psi(x)=(\psi_{1}, ..., \psi_{8})^T$ each of $\psi_{i}$ a fermion or Dirac spinor, with four complex components, i.e. spinor indices suppressed, we can write for example $L_{1}\psi(x)$ to make or show a transformation of the fermion fields.

Now my question is that in the example of $SU(3)$ are these two different Adjoint Representations related somehow, as one of them involves the operation of $3*3$ matrices and the another one involves the operation of $8*8$ matrices?

My first impression is that in the first case, the 3 dimensional one, we are using the adjoint representation of the group using the Lie algebra as a vector space whereas in the second case, the 8 dimensional one, we are using the adjoint representation of the Lie algebra using the Lie algebra itself as a vector space. And both of them are useful in writing the Lagrangian density of the SM. Is this correct?