Please can you help me as I have just been introduced to your Help Forum:
Determine whether the vectors u, v and w given below are linearly independent or dependent where u, v and w are non-colliner vectors such that
u=2a-3b+c , v=3a-5b+2c and w=4a-5b+c.
thanks.
Use exactly the same method I just showed "Babanla"
set them as a matrix 3x3 then find the determinant if it is 0 then they are dependant
To determine whether the vectors u, v, and w are linearly independent or dependent, we need to examine the relationship between these vectors. In other words, we need to check if there exist any scalars (constants) k1, k2, and k3, not all zero, such that k1u + k2v + k3w = 0.
Here's how you can check if the vectors are linearly dependent or independent:
1. Create an equation using the given vectors. Use the coefficients k1, k2, and k3 as variables:
k1u + k2v + k3w = 0
2. Plug in the expressions for u, v, and w using the given values:
k1(2a-3b+c) + k2(3a-5b+2c) + k3(4a-5b+c) = 0
3. Distribute the k values and collect like terms:
(2k1+3k2+4k3)a + (-3k1-5k2-5k3)b + (k1+2k2+k3)c = 0
4. By comparing the coefficients of the variables a, b, and c, form a system of linear equations:
2k1+3k2+4k3 = 0 (equation 1)
-3k1-5k2-5k3 = 0 (equation 2)
k1+2k2+k3 = 0 (equation 3)
5. Solve the system of equations using any method you prefer, such as substitution or elimination.
Once you have the solution for the values of k1, k2, and k3, you can determine if the vectors u, v, and w are linearly independent or dependent.
If the only solution to the system of equations is k1 = k2 = k3 = 0, then the vectors u, v, and w are linearly independent. This means that none of the vectors can be written as a linear combination of the others.
If there exists a solution where at least one k value is not zero, then the vectors u, v, and w are linearly dependent, indicating that one or more of the vectors can be written as a linear combination of the others.