Linear algebra, a branch of mathematics that deals with linear equations, linear transformations, and the algebraic structures of vector spaces, involves the study of vectors and matrices, using tools such as addition, scalar multiplication, and linear transformations to analyze systems of linear equations, and understand the relationships between vectors and matrices; beginning with vectors, which are ordered lists of numbers, or elements, represented as rows or columns, and can be added or subtracted element-wise, i.e., (a1, a2, …, an) + (b1, b2, …, bn) = (a1+b1, a2+b2, …, an+bn) and scaled by a scalar (constant) c, i.e., c(a1, a2, …, an) = (ca1, ca2, …, can), we then explore vector spaces, which are sets of vectors that are closed under addition and scalar multiplication, satisfying eight axioms (associativity, commutativity, existence of identity and inverse elements, and compatibility of scalar multiplication with field operations); furthermore, we define a subspace as a subset of a vector space that is also a vector space itself, and a linear combination as a sum of scalar multiples of vectors, which leads to the concept of span, which is the set of all linear combinations of a given set of vectors, and linear independence, where no vector in a set can be expressed as a linear combination of the others, with a basis being a linearly independent set that spans the vector space, and the dimension being the number of vectors in a basis, while we can also project a vector onto another vector, with the projection of u onto v being proj_v(u) = ((u⋅v) / (||v||^2))v, where u⋅v is the dot product, or inner product, of u and v, defined as u⋅v = u1v1 + u2v2 + … + unvn, and ||v|| is the norm, or length, of v, calculated as √(v1^2 + v2^2 + … + vn^2); proceeding to matrices, which are rectangular arrays of numbers organized into rows and columns, where the entry a_ij is the element in the i-th row and j-th column, we can perform matrix addition and subtraction element-wise, i.e., A + B = (a_ij + b_ij), and matrix multiplication, i.e., C = AB, with c_ij = a_i1b_1j + a_i2b_2j + … + a_inb_nj, where A has dimensions m×n and B has dimensions n×p, resulting in C having dimensions m×p; moreover, we can multiply a matrix by a scalar, i.e., cA = (ca_ij), and calculate the transpose of a matrix, i.e., A^T = (a_ji), as well as the determinant of a square matrix (n×n), using the recursive definition det(A) = Σ((-1)^(i+j)a_ijM_ij), where M_ij is the (i, j) minor, or the determinant of the submatrix obtained by deleting the i-th row and j-th column, with the determinant having essential properties, such as det(AB) = det(A)det(B) and det(A^T) = det(A), and being useful for finding the inverse of a matrix, A^(-1), which exists only if det(A) ≠ 0 and satisfies AA^(-1) = A^(-1)A = I, where I is the identity matrix, having ones on the diagonal and zeros elsewhere; additionally, linear algebra employs matrix row operations, including row swapping, row scaling, and row addition, to transform a given matrix into an equivalent matrix with a simpler form, such as the row-echelon form or the reduced row-echelon form, which aids in solving systems of linear equations, represented as Ax = b, where A is the coefficient matrix, x is the unknown vector, and b is the constant vector, and we can apply the Gaussian elimination method to find the unique solution, infinitely many solutions, or no solution depending on the nature of the system; furthermore, we explore linear transformations, which are functions that map vectors from one vector space to another while preserving vector addition and scalar multiplication, i.e., T(u + v) = T(u) + T(v) and T(cu) = cT(u), and we can represent linear transformations using matrices, with the transformation T being represented by the matrix A, such that T(x) = Ax, and can analyze properties like injectivity (one-to-one), surjectivity (onto), and bijectivity (both injective and surjective), as well as the kernel (null space) and image (column space) of a transformation, defined as ker(T) = {x | T(x) = 0} and im(T) = {T(x) | x ∈ V}, respectively, with the rank-nullity theorem stating that dim(ker(T)) + dim(im(T)) = dim(V), where V is the domain of the transformation; finally, linear algebra encompasses eigenvalues and eigenvectors, which are scalar-vector pairs (λ, v) satisfying Av = λv, where A is a square matrix, and are crucial in many applications, including diagonalization of matrices, where A is diagonalizable if it can be written as A = PDP^(-1), with D being a diagonal matrix containing the eigenvalues on the diagonal, and P being a matrix with eigenvectors as columns, as well as singular value decomposition, a generalization of diagonalization for non-square matrices, and other advanced topics like orthogonal and unitary matrices, Hermitian and symmetric matrices, positive definite matrices, and quadratic forms, which have important implications in various fields such as physics, engineering, computer science, and data science, all of which make linear algebra a fundamental and versatile area of mathematics with numerous applications and deep connections to other branches of mathematics, like geometry, calculus, and abstract algebra.