For each set of vectors in parts a b and c indicate whether

For each set of vectors in parts (a), (b) and (c), indicate whether all linear combinations lie on a line plane or all of R^3 [1 2 3] and [3 6 9] [1 0 0] and [0 2 3] [2 0 0] and [0 2 2] and [2 2 3] For coordinate vectors vector i, vector j in xyz (R^3) space, describe in terms of x, y and/or z the plane of all linear combinations of vector i = (1, 0, 0) and vector i + vector j = (1, 1, 0). How long is the vector V = (1, 1, ..., 1) in 6 dimensions? Find a unit vector vector v in the same direction as V and a unit vector^W that is perpendicular to v.

Solution

a> dimension of R³ =3 and there two dependent vectors as 2nd vector is a scalar multiple of the 1st. So all linear combination won\'t lie on the plane

b> dimension of R³=3 and to generate R³, a set containing 3 independent vectors is required . Here the given vectors are independent but there are only 2 vectors. So it won\'t be possible

c>dimension of R³ =3 and the set contains 3 linearly independent vectors , so all possible linear combinations will lie on a plane