MHB Rank & Nullity: 3x3 Matrix w/ Plane Origin & LD Vectors

[Swati]

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]

#2 rowrank(A) = columnrank(A) -- seems to follow immediately.

 

[Swati]

we have to prove. please explain clearly.

 

[caffeinemachine]

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]

we have to prove. please explain clearly.

More effort on your part is required. What have you tried and what don't you understand.