Choose the correct Answers Given the following vector set u1

Choose the correct Answer(s):

Given the following vector set: u=[1,2,3,4], v=[-1,-2,3,4], w=[0,1,1,0]

What is the dimension of the space that these vectors reside in and is this space spanned by the set?

1- Dimension is 3 and the space is not spanned by the set
2- Dimension is 4 and the space is spanned by the set
3- None of these
4- Dimension is 3 and the space is spanned by the set
5- Dimension is 2 and the space is spanned by the set
6- Dimension is 4 and the space is not spanned by the set

Solution

Let A be the matrix with the given vectors as columns. Then the column vectors of the RREF of A are ( 1, 0,0,0)^T , ( 0,1,0,0)^T and ( 0,0,0,1)^T . This means that the given vectors are linearly independent and span R⁴.These vectors form a basis for R⁴. The dimension is 3. Thus the statement number 4 is correct. The dimension is 3 and the space is spanned by the set.