Let A be a 4x7 matrix with rank 4. Which of the following(s) is/are true?
1) the rows of A are linearly independent
2) the columns of A must be linearly dependent
4) the null space of A must have dimension 3
8) if we delete any three columns of A, the resulting matrix must be invertible

Question

Let A be a 4x7 matrix with rank 4. Which of the following(s) is/are true?
1) the rows of A are linearly independent
2) the columns of A must be linearly dependent
4) the null space of A must have dimension 3
8) if we delete any three columns of A, the resulting matrix must be invertible

anonymous · Answer

I think 1, 2 are correct, 8 is not. Not sure about 4.

anonymous · Answer

4 should be right, I guess
rank(A) + dim (null(A)) = 7
dim (null(A)) = 7-4 = 3

experimentx · Answer

yes one and two are correct ...

anonymous · Answer

But 4 and 8?

experimentx · Answer

third is also correct ... null space + row space  = 7

experimentx · Answer

the last is not correct

anonymous · Answer

I would have lost 2 points in exam since I was soooooo careless lol

anonymous · Answer

Thanks for checking :D

experimentx · Answer

any three columns of A ... is incorrect. make 000 for three columns out of 7 ... and delete other columns, still the matrix is not invertible.

experimentx · Answer

some columns would have been right.

anonymous · Answer

I know 8 lol. I got it wrong for 1 and 4 but after checking (wiki) for 4 and think about 1, then got the right answers lol

experimentx · Answer

that's very good (y)