Linear Algebra and Vector Equations – Math Essay
Week 2 in Linear Algebra focused on Vector Equations (Section 1.3) and the Matrix Equation Ax = b (Section 1.4). The key concepts in section 1.3 are linear combinations of vectors and a spanning set. A vector is

defined as a matrix with one column. A vector with n entries defines a vector in Rn. If you plot two vectors in R2, you can find the sum of the vectors graphically using Parallelogram Rule. Linear Combinations are combinations of vectors, using different weights for each. For example, given vectors v and u, 2v + u is a Linear Combination of the two vectors. Furthermore, the equation v + u = b has the same solution as the augmented matrix [v u | b]. This means that b can be made from a linear combination of v and u if and only if the corresponding augmented matrix has a solution. The other key concept in section 1.3 is Span. The Span of vectors is the set of all linear combinations of the vectors. Given u, v, and z, Span{ u, v, z} is the collection of all vectors that can be written as x1u + x2v + x3z. To determine if vector b is in the span of vectors u and v, put all three vectors into an augmented matrix (vector b being the last row) and solve. If the solution is consistent, b is in the Span{u, v}.

Section 1.4 explains how linear combinations can be viewed as vector multiplication. There are three ways of viewing linear systems: as a system of linear equations, as a vector equation, or as a matrix equation Ax = b. So, since all three represent linear systems, they can all be used to find the same solutions. For any matrix A with rows m and columns n, the following is true: for each b in Rm, the equation Ax = b has a solution, each b in Rm is a linear combination of the columns of A, the columns of A span Rm, and A has a pivot position in every row. This means that a 3×3 matrix must have a pivot in every row to span R3. Finally, the last theorem of the section states that if A is an m x n matrix, u and v are vectors in Rn, and c is a scalar, then: A(u+v) = Au + Av, and A(cu) = cAu.