
#1
Mar2310, 07:16 PM

P: 15

If n*n matrix, can row space ever be equal to null space?
P.S.: this is NOT a homework question. It's a general question to get the concepts straight in my head. 



#2
Mar2310, 07:57 PM

P: 403

Suppose v is a vector that belongs to the row and null spaces of A, then:
[tex]v^T = a^{T}A[/tex] And: [tex]Av=\bold 0[/tex] Therefore, [itex]v^{T}v[/itex] must be equal to what? And what does this imply regarding v? 



#3
Mar2410, 12:29 PM

P: 15





#4
Mar2410, 01:34 PM

P: 403

If n*n matrix, can row space ever be equal to null space?
The transpose, of course.




#5
Apr210, 12:33 AM

P: 118

The nullspace and rowspace of A are complementary subspaces in R^n (i.e. dim of nullspace + dim of rowspace = n and the dot of any vector in the nullspace with any vector in the rowspace will give 0)
vectors in the basis of the nullspace are of the form Ax=0 vectors in the basis of the rowspace are the pivots rows of the row reduced echelon form of A Consider the zero Matrix 



#6
Apr710, 02:27 PM

P: 118

Any invertible matrix has a rowspace that spans R^n and it's nullspace will be the zero vector Any singular matrix will have k vectors in its rowspace basis and z vectors in its nullspace basis such that z + k = n since the only vector dotted with itself that gives zero is the zero vector then the nullspace and rowspace would have to both be equal to the zero vector to give zero, meaning that the matrix must be "empty" for that to be possible and an "empty" matrix isn't a matrix at all, so there exists no matrix such that nullspace = rowspace 


Register to reply 
Related Discussions  
Null space of 3x3 matrix  Linear & Abstract Algebra  0  
Finding null space of a given matrix  Calculus & Beyond Homework  8  
Finding the null space, matrix fun! wee!  Calculus & Beyond Homework  5  
Linear Algebra: Basis for the Null Space of a Matrix  Calculus & Beyond Homework  5  
Question about the Null Space for this Zero Matrix  Introductory Physics Homework  5 