Does this set of vectors span all of 4-space?

Question

Determine whether the set of four vectors below span all of 4-space:
[1 0 0 3], [0 3 1 2], [0 3 0 2], [0 0 1 0]

Please show your work because I am more interested in the method than the answer (but I would still like the answer). Thank you.

Answer 1

No. Check the rank of the matrix. If it is less than 4 then the vectors do not span all of 4-space.

1 0 0 3
0 3 1 2
0 3 0 2
0 0 1 0

Subtract the third row from the second row.

1 0 0 3
0 0 1 0
0 3 0 2
0 0 1 0

Subtract the second row from the fourth row.

1 0 0 3
0 0 1 0
0 3 0 2
0 0 0 0

The rank of the matrix is less than 4 so the set of vectors does NOT span 4-space.

Answer 2

the easiest way is to set up the vectors as a matrix like
[1 0 0 0]
[0 3 3 0]
[0 1 0 1]
[3 2 2 0]
and then row reduce it to
[ * * * *]
[0 * * *]
[0 0 * *]
[0 0 0 *]
where the stars are just whatever numbers. if theres a pivot position in every row ( i.e. there is not a row of all zeros) then the column vectors span 4 space

Answer 3

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