Final answer:
To determine whether the given vectors are linearly independent or dependent in R4, set up a system of equations and solve for the scalars. If the only solution is the trivial solution, the vectors are linearly independent. If a non-trivial solution exists, the vectors are linearly dependent.
Step-by-step explanation:
Two vectors in ℝ4 are linearly dependent if one of the vectors can be written as a linear combination of the other vectors. In other words, if you can find scalars (real numbers) such that the linear combination equals zero, then the vectors are linearly dependent. Otherwise, they are linearly independent.
To determine whether the given vectors are linearly independent or dependent, you can set up a system of equations and solve for the scalars. If you find a non-trivial solution (where not all scalars are zero), then the vectors are linearly dependent. If the only solution is the trivial solution (where all scalars are zero), then the vectors are linearly independent.
For example, suppose you have two vectors a = [1, 2, 3, 4] and b = [2, 4, 6, 8]. You can set up the system of equations 1a + 2b = 0 and solve for the scalars. If you find a non-trivial solution, such as a = 2 and b = -1, then the vectors are linearly dependent. If the only solution is the trivial solution, such as a = 0 and b = 0, then the vectors are linearly independent.