# Column Space, Row Space, Null Space, and Left Null Space

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!

• 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. – R zu Nov 4 '18 at 4:49
• 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. – 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.