Rank and Nullity

Swati

New member
1.(a) Give an example of 3*3 matrix whose column space is a plane through the origin in 3-space
(b) what kind of geometry object is the null space and row space of your matrix

2. Prove that if a matrix A is not square, then either the row vectors or the column vectors of A are linearly dependent.

tkhunny

Well-known member
MHB Math Helper
#2 rowrank(A) = columnrank(A) -- seems to follow immediately.

Swati

New member
we have to prove. please explain clearly.

caffeinemachine

Well-known member
MHB Math Scholar
1.(a) Give an example of 3*3 matrix whose column space is a plane through the origin in 3-space
(b) what kind of geometry object is the null space and row space of your matrix

2. Prove that if a matrix A is not square, then either the row vectors or the column vectors of A are linearly dependent.
I assume that the entries of the matrices are reals.
Q2. Consider a 2x3 matrix. Let its row vectors be (a,b), (c,d),(e,f). So we got 3 vectors from $\mathbb{R}^2$. They have to be linearly dependent since dimension of $\mathbb{R}^2$ is 2. Can you generalize?

Mr Fantastic

Member
we have to prove. please explain clearly.
More effort on your part is required. What have you tried and what don't you understand.