Final answer:
To determine if every vector in R⁴ can be written as a linear combination of the columns of a matrix B, we need to check if the columns of B span R⁴.
Step-by-step explanation:
In order to determine whether every vector in R⁴ can be written as a linear combination of the columns of a given matrix B, we need to check if the columns of B span the entire vector space R⁴. If the columns of B span R⁴, then any vector in R⁴ can be expressed as a linear combination of those columns. One way to check if the columns of a matrix span a vector space is to row reduce the matrix and look for a row of all zeros, which would indicate that some vector in R⁴ cannot be expressed as a linear combination of the columns. If there are no rows of all zeros, then the columns span the entire vector space and every vector in R⁴ can be written as a linear combination of the columns of B.