I'm a bit confused on the relationship between Column Space, Row Space, Null Space, and Left Null Space.

I know that by the rank-nullity theorem, column space cannot equal null space. Does this also mean that the column space cannot equal the left null space?

I think that column space can equal row space, but I am not sure. I also do not know if the row space can equal the null space.

This would be for a $3 \times 3$ matrix.

Any help would be greatly appreciated! Thank you!

  • $\begingroup$ If the rows and columns are identical (as in a symmetric matrix), row space and column space are definitely the same. Don't know if there are other cases. $\endgroup$ – R zu Nov 4 '18 at 4:49
  • $\begingroup$ If the eigenvalues of a n by n square matrix are all distinct, I guess all columns form a basis for $R^{n}$. Same for the rows. Then, the row space and column space are the same. $\endgroup$ – R zu Nov 4 '18 at 5:02

A matrix is not just as an array of numbers. It is helpful to think of it as a device for takes a vector as input and produces another vector as output by multiplication: that is for input $v$, the output is $Av$. This output is obtained by taking linear combination of column vectors of $A$, the coefficients for the linear combination being provided by the components of the vector $v$. So the output belongs to the column space.

It is possible that $Av$ is the zero vector,in that case $v$ is said to be in the nullspace.

For left multiplication $vA$, again one has similar interpretation, but everything in terms of rows of $A$ instead of the columns.

Now look at a matrix like $A=\pmatrix{1 & 2 & 3\cr 1 & 2 & 3 \cr 1 & 2 & 3\cr}$. Any scalar multiple every column vector is of the form $(x, x, x)^T$, and so linear combination would also be of the same type. So column space consists exclusively of vectors of the kind $(x,x,x)^T$. But any vector in the row space of $A$ is clearly of the form $(y, 2y, 3y)^T$. So column space and row space have nothing in common except the zero vector.

When the rank is 3, the columns form a basis for $\mathbf{R}^3$, and so every vector is in the column space, including those in the row space, and vice versa.

When the matrix is symmetric then also we cna check row space and column spaces coincide. Ohterwise they don't. Only thing we can say is those spaces have the *same dimensions$ which is much different from saying they are the same.


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