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How do you find the dimension of the subspace of R4 consisting of the vectors a plus 2b plus c b-2c 2a plus 2b plus c 3a plus 5b plus c?
The dimension of a space is defined as the number of vectors in its basis. Assuming your vectors are 1,2,1,0 0,1,-2,0 2,2,1,0 and 3,5,1,0 (extra zeros because you are in R4) then you must first check to see if they are linearly indepent. If all the vectors are linearly independent then the subspace defined by those vectors has a dimension 4, as there are 4 vectors in the basis.
What is coplanar vector?
In geometry a vector is used to make the equations easier to understand and to figure out. Velocity and force are examples of vectors. For a vector to be coplanar there must be two or more and they must be linearly dependent. Coplanar vectors have proportional components and their rank is 2.
What is the condition for being 3 vectors in a plane?
The general idea is that 3 vectors are in a plane iff they are not linearly independent. This can be checked in several ways:guessing a way to represent one of them as a linear combination of the other two - if it can be done, then they are coplanar;if they are three-dimensional, simply by calculating the determinant of the matrix whose columns are the vectors - if it's zero, they are coplanar, otherwise, they aren't;otherwise, you may calculate the determinant of their gramian matrix, that is, a matrix whose ij-th entry is the dot product if the i-th and j-th of the three vectors (e.g. it's 1-2-nd entry would be the dot product of first and second of them); they are coplanar iff the determinant is zero.
Is every bijection a strictly monotonic function?
No. For example, consider the discontinuous bijection that increases linearly from [0,0] to [1,1], decreases linearly from (1,2) to (2,1), increases linearly from [2,2] to [3,3], decreases linearly from (3,4) to (4,3), etc.
What is arthropod vectors?
Vectors of the arthropod.
