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Rank and Nullity

Swati

New member
Oct 30, 2012
16
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
Jan 27, 2012
267
#2 rowrank(A) = columnrank(A) -- seems to follow immediately.
 

Swati

New member
Oct 30, 2012
16
we have to prove. please explain clearly.
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
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
Jan 26, 2012
66
we have to prove. please explain clearly.
More effort on your part is required. What have you tried and what don't you understand.