determine if the following vectors are linearly independent or dependent u=(-4,5) v=(2,7)

To determine if the vectors u=(-4,5) and v=(2,7) are linearly independent or dependent, we can check if one vector can be expressed as a linear combination of the other vector. Let's assume that the vectors u and v are linearly dependent. This means that there exist scalars a and b, not both zero, such that: au + bv = 0 Substituting the values of u and v: a(-4,5) + b(2,7) = (0,0) Simplifying the equation: Simplifying the equation: (-4a + 2b, 5a + 7b) = (0,0) This gives us the following system of equations: -4a + 2b = 0 5a + 7b = 0 To determine if this system has a nontrivial solution (a solution where at least one of the scalars is not zero), we can solve the system of equations. Using any method of solving linear equations, we find that the only solution to this system is a = 0 and b = 0. This means that the only way to satisfy the equation au + bv = 0 is if both a and b are zero. Since the only solution to the equation is the trivial solution, we can conclude that the vectors u=(-4,5) and v=(2,7) are linearly independent. Therefore, the vectors u=(-4,5) and v=(2,7) are linearly independent. determine if the following vectors are linearly independent or dependent u=(-4,5) v=(2,7) Quick Search youtube video preview thumbnail Answer The vectors u=(-4,5) and v=(2,7) can be checked for linear independence or dependence by assuming that they are linearly dependent and checking if one vector can be expressed as a linear combination of the other vector. If the only solution to the equation au + bv = 0 is a = b = 0, then the vectors are linearly independent. Here are the steps to determine if the vectors are linearly independent: Assume that the vectors u and v are linearly dependent. Find scalars a and b, not both zero, such that au + bv = 0. Substitute the values of u and v into the equation and simplify. Solve the resulting system of equations to find the values of a and b. If the only solution to the equation is a = b = 0, then the vectors are linearly independent. Otherwise, they are linearly dependent. Using this method, we find that the vectors u=(-4,5) and v=(2,7) are linearly independent, since the only solution to the equation au + bv = 0 is a = b = 0. Therefore, the vectors u=(-4,5) and v=(2,7) are linearly independent. Related