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

At the graduate level, linear algebra shifts from the calculation of matrices to the study of formal algebraic structures. The foundational structure is the **vector space**, a set of objects called vectors that can be added together and multiplied by scalars. While undergraduate courses often focus on Euclidean space, graduate study treats vector spaces as abstract entities defined over a **field**. ### Fields and Scalars A field is a set of scalars—such as the real numbers, complex numbers, or finite fields—where addition, subtraction, multiplication, and division (except by zero) are well-defined and follow standard commutative and associative laws. In abstract linear algebra, a vector space is defined "over" a specific field, meaning the scalars used for multiplication must belong to that field. This abstraction allows the theory to apply to functions, polynomials, and cryptography beyond simple geometric vectors. ### Subspaces A **subspace** is a subset of a vector space that remains a vector space under the original operations of addition and scalar multiplication. For a subset to qualify as a subspace, it must satisfy three conditions: 1. It must contain the zero vector. 2. It must be closed under addition (the sum of two elements is in the subset). 3. It must be closed under scalar multiplication (a scalar times an element is in the subset). ### Linear Independence and Spanning Sets The structure of a vector space is determined by how its elements relate to one another through linear combinations. - **Linear Independence**: A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others. Formally, this means the only way to reach the zero vector using a linear combination of these vectors is to set all scalars to zero. - **Spanning Set**: A set of vectors spans a space if every vector in that space can be expressed as a linear combination of the vectors in the set. The **span** represents the collection of all possible linear combinations. ### Basis and Dimension The concepts of independence and spanning converge in the definition of a **basis**. A basis is a set of vectors that is both linearly independent and spans the entire vector space. A basis provides a unique "coordinate system" for the space, as every vector can be represented by exactly one unique linear combination of basis vectors. The **dimension** of a vector space is defined as the number of vectors in its basis. This is an intrinsic property of the space; while a space may have infinitely many different bases, every basis for a specific space will always contain the same number of elements. Graduate study often explores both finite-dimensional spaces and infinite-dimensional spaces, such as those found in functional analysis.