Finding x: When is v in the Span of a, b, c, d?

In linear algebra, the span of a set of vectors represents all possible vectors we can create through linear combinations.Let's start with two vectors, a and b. We can multiply each vector by any scalar value.When we multiply a vector by a scalar, we change its length while keeping its direction.A linear combination is when we multiply each vector by a scalar and add the results together.Here's how we add vectors: First multiply each vector by its scalar, then combine them tip-to-tail.The resulting vector is part of the span of vectors a and b.The span includes every possible vector we can create through linear combinations of our original vectors.Any point in this shaded region can be reached by some combination of our vectors.Mathematically, we write the span as the set of all vectors that can be created using real number scalars s1 and s2.To determine if a vector v is in the span, we need to express it as a linear combination of our basis vectors.Let's understand what each component represents.We can rewrite this equation in matrix form, where our vectors become columns of a matrix A.Matrix A contains our basis vectors as columns, x contains our scalar values, and together they must equal vector v.This can be written more compactly as A x equals v.To solve this system, we can write it as an augmented matrix, combining A and v with a vertical line.Here's an example with actual numbers. Each row represents one equation in our system.This augmented matrix form lets us solve the system using row operations in the next step.Now we're ready to solve this system using Gaussian elimination.Now that we have our augmented matrix, let's solve it using Gaussian elimination.First, we'll use the first row as a pivot to eliminate entries below it.Next, we'll use the second row to eliminate entries below it.Finally, we get our reduced row echelon form.From our reduced matrix, we can read off the solution. In this case, we have a unique solution.When solving these systems, we can encounter three different scenarios.Geometrically, when we have a unique solution, our vector v lies exactly on a point in the span.If we have infinite solutions, v lies in a region where multiple combinations of our vectors can reach it.And if we have no solution, v lies outside the span entirely.