I want to study Linear Algebra to a graduate level what are the key topics

In advanced mathematics, the study of a vector space $V$ over a field $F$ is complemented by the analysis of its **dual space**, denoted as $V^*$. This study is fundamental for understanding tensors, differential geometry, and functional analysis. ### Linear Functionals and the Dual Space The dual space $V^*$ is defined as the set of all **linear functionals** on $V$. A linear functional is a linear transformation $f: V \to F$ that maps vectors to scalars. If $V$ is finite-dimensional, $V^*$ is also a vector space of the same dimension. While $V$ and $V^*$ are isomorphic, they represent fundamentally different objects: $V$ contains vectors, while $V^*$ contains operators that act upon those vectors. ### Dual Bases For a finite-dimensional vector space $V$ with a basis $\{v_1, \dots, v_n\}$, there exists a unique **dual basis** $\{f_1, \dots, f_n\}$ in $V^*$. This basis is defined by the property: - $f_i(v_j) = 1$ if $i = j$ - $f_i(v_j) = 0$ if $i \neq j$ This relationship allows any linear functional to be expressed as a linear combination of these basis functionals. The dual basis is a critical tool in change-of-basis problems and the representation of linear operators as matrices. ### Annihilators Given a subspace $W$ of $V$, the **annihilator** $W^0$ is the set of all linear functionals in $V^*$ that map every vector in $W$ to zero. The annihilator is itself a subspace of $V^*$. A key theorem in graduate linear algebra relates the dimensions of these spaces: 1. The dimension of $W$ plus the dimension of $W^0$ equals the dimension of $V$. 2. Annihilators provide a dual perspective on systems of linear equations, where the solution space is the kernel of the functionals defining the equations. ### The Double Dual and Reflexivity The **double dual** $V^{**}$ is the dual space of $V^*$. In finite-dimensional contexts, there is a **canonical isomorphism** between $V$ and $V^{**}$. This means every vector $v$ in $V$ can be viewed as a functional on $V^*$ through an evaluation map: $\hat{v}(f) = f(v)$ for any $f \in V^*$. This property is called **reflexivity**. Unlike the isomorphism between $V$ and $V^*$, which depends on the choice of basis, the isomorphism between $V$ and $V^{**}$ is "natural" because it is defined without reference to a specific basis. In infinite-dimensional spaces, this reflexivity is not guaranteed, making the double dual a central topic in the study of Banach and Hilbert spaces.