# Rank of a matrix

Hi
I don't understand why only a matrix full of zero has a rank = 0.

"the rank of a matrix A is the number of linearly independent rows or columns of A"

If I have a 3x3 matrix

A = [ 1 1 1
1 1 1
1 1 1 ]

assuming a_i denotes the column or row vector i of A. I can say

a_1 = 1*a_2 + 0*a_3 so a_1 is not linearly independant
a_2 = 1*a_1 + 0*a_3 so a_2 is not linearly independant
a_3 = 1*a_1 + 0*a_2 so a_3 is not linearly independant

So why rank A = 1 and not 0 ?
I know i'm missing something, I don't know what!

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lurflurf
Homework Helper
To have rank 0 Ax=0 for all x. The column (or row vectors) are linearly dependent in pair and triples, but linearly dependent in singles. The rank is r if there exist r rows or columns that are linearly independent.

rank is the dimension of the subspace composed by the set of points you can reach using constant multiples of the vectors in your matrix.
A 3x3 matrix of ones can reach any point on a line in R3 (which is a subspace) and lines have dimension 1, so rank is 1.

HallsofIvy
Homework Helper
More abstractly, an n by n matrix represents a linear transformation from an n dimensional vector space to an n dimensional vector space- L: U-> V. The "range" is the dimension of L(U) as a subspace of V. In particular, if you multiply a matrix by the vector having 1 as the ith entry, 0 every where else, you get the ith column of the matrix. But the set of all such vectors form a basis for U and so are mapped into a set that spans L(U). The only subspace with dimension 0 is the set containing only the 0 vector. In other words, to have rank 0, L must map every vector into the 0 vector. That is the "0" linear tranformation which is represented by the 0 matrix.

Deveno