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What is an intuitive meaning of the null space of a matrix? Why is it useful?

I'm not looking for textbook definitions... my textbook gives me the definition, but I just don't "get" it.

E.g.: I think of the rank R of a matrix as the minimum number of dimensions that a linear combination of its columns would have; it tells me that, if I combined the vectors in its columns in some order, I'd get a set of coordinates for an R-dimensional space, where R is minimum (please correct me if I'm wrong). So that means I can relate rank (and also dimension) to actual coordinate systems, and so it makes sense to me. But I can't think of any physical meaning for a null space... could someone explain what its meaning would be, for example, in a coordinate system?

Thanks!

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Your statement "the rank R of a matrix as the minimum number of dimensions that a linear combination of its columns would have..." should be "the rank R of a matrix as the maximum number of dimensions that a linear combination of its columns would have...". The rank tells you the dimension of a space spanned by the columns. –  Tpofofn Feb 11 '11 at 2:01

3 Answers 3

up vote 13 down vote accepted

If $A$ is your matrix, the null-space is simply put, the set of all vectors $v$ such that $A \cdot v = 0$. It's good to think of the matrix as a linear transformation; if you let $h(v) = A \cdot v$, then the null-space is again the set of all vectors that are sent to the zero vector by $h$. Think of this as the set of vectors that lose their identity as $h$ is applied to them.

Note that the null-space is equivalently the set of solutions to the homogeneous equation $A \cdot v = 0$.

Nullity is the complement to the rank of a matrix. They are both really important; here is a similar question on the rank of a matrix, you can find some nice answers why there.

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Ohhhhhhhhhhhhhhhhhhhhhhhh the "loses their identity" part made so much sense! So that is why, when we reduce the dimensions of an m * n matrix, the number of vectors that don't lose their identity (the number of pivot columns) + the number of vectors that do (which is dim Null A) is just the total number of columns, n... thanks! It makes so much more sense now! :) –  Mehrdad Feb 9 '11 at 7:41

The rank $r$ of a matrix $A \in \mathbb{R}^{m \times n}$, as you have said is the dimension of the column space ($r$ is also the dimension of the row space as well) i.e. the dimension of the space spanned by vectors which are obtained by a linear combination of the columns of $A$, equivalently the range of $A$. (The use of the word "minimum" in the question is unnecessary). However each column vector has $m$ components and the vectors in the range of $A$ has $m$ components as such but span only a $r (\leq m)$ dimensional subspace instead of a $m$ dimensional space. So we are missing out spanning the remaining $m-r$ dimensional subspace of the $m$ dimensional space.

The left null-space now plays the roll of spanning the remaining $m-r$ dimensional subspace. This is why the left null-space is orthogonal to the column space. So the left null-space along with the column space now spans the entire $m$ dimensional space i.e. if $C = \{y \in \mathbb{R}^{m \times 1}: y = Ax\text{ for some }x \in \mathbb{R}^{n \times 1} \}$ and $Z_L = \{z \in \mathbb{R}^{m \times 1}:z^T A = 0 \}$,

then $Z_L \cup C = \mathbb{R}^{m}$ and $Z_L \perp C$

The right null-space plays the analogous roll for the rows. The rows span only a $r$ dimensional subspace of the $n$ dimensional space. The right null-space now plays the roll of spanning the remaining $n-r$ dimensional subspace. This is why the right null-space is orthogonal to the row space. So the right null-space along with the row space now spans the entire $n$ dimensional space i.e. if $R = \{y \in \mathbb{R}^{n \times 1}: y = A^Tx\text{ for some }x \in \mathbb{R}^{m \times 1} \}$ and $Z_R = \{z \in \mathbb{R}^{n \times 1}: Az = 0 \}$,

then $Z_R \cup R = \mathbb{R}^{n}$ and $Z_R \perp R$

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This is a good mathematical explanation, but it's not really intuitive for me. (It's probably just me, not your explanation.) So far, I've tended to think of linear algebra as a tool for figuring out the number of independent variables (slash, coordinates) in an equation (or matrix), so putting it in terms of that would be more intuitive to me than just a purely mathematical definition of rows and columns. It makes me intuitively see answers without worrying about vocabulary. But +1, nice explanation anyhow. :) –  Mehrdad Feb 9 '11 at 7:47

Imagine a set of map directions at the entrance to a forest. You can apply the directions to different combinations of trails. Some trail combinations will lead you back to the entrance. They are the null space of the map directions.

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