Two vectors are considered linearly independent if there is no way to write one of the vectors as a scalar multiple of the other. In other words, they do not lie on the same line and are not redundant in expressing a given vector space.
To prove that two vectors, Au and Av, are linearly independent, you can use the definition of linear independence. This means that you need to show that the only solution to the equation cAu + dAv = 0 (where c and d are scalars) is when c = d = 0. You can do this by setting up a system of equations and solving for c and d.
Yes, using a system of equations is a common method for proving linear independence. This method involves setting up a system of equations with the two vectors and their corresponding coefficients. If the only solution to this system is c = d = 0, then the vectors are linearly independent.
Geometrically, linear independence means that the two vectors do not lie on the same line. This can also be interpreted as the vectors pointing in different directions and not being redundant in expressing a given vector space.
Yes, it is possible to have more than two linearly independent vectors. In fact, the number of linearly independent vectors in a given vector space is known as the dimension of that space. This means that a higher number of linearly independent vectors can span a higher-dimensional vector space.