Finding the Span of Two Vectors in R4

Find the span of these two vectors in $\mathbb{R}^4$ using the row reduction technique to determine the dimension.

In this problem, you are tasked with finding the span of two vectors in the four-dimensional space, R4. A vector span refers to the set of all linear combinations of a given set of vectors. In simpler terms, it is the collection of all vectors that can be formed by scaling and adding the given vectors.

Understanding the concept of a span is crucial in linear algebra as it helps in determining the coverage or reach of given vectors within a vector space. To solve this problem, we use the method of row reduction, also known as Gaussian elimination. Row reduction is a pivotal technique in linear algebra, often employed to solve systems of linear equations, find matrix inverses, and determine ranks and dependencies among vectors.

The process involves performing a series of operations to transform a matrix into its row echelon form or even further to its reduced row echelon form. By doing so, we can glean important insights about the relationships between the vectors, such as checking for linear independence.

Linear independence is a key concept here. If the vectors are linearly independent, their span will form a plane in R4, with the dimension equal to 2. However, if the vectors are linearly dependent (one is a scalar multiple of the other), the span will merely be a line, with the dimension equal to 1.

Thus, through row reduction and understanding the definition of spans, you can determine both the span and dimension of the given vectors in R4.