Based on the following axioms of vector spaces where u & v are vectors and c is a scalar:
• u + v is in V
• c*u is in V
- I thought that the "space" of a specific matrix's vector space was all the linear combinations of its vectors, but I later found that to be the column space of that particular matrix.
If Null and Column spaces are subspaces of a Matrix's vector space, then what exactly makes up the Vector Space of a matrix? Is it the set of vectors as a whole?
Also, the definition of a subspace states that it is a "subset" of vectors from a larger vector space. The column Space seems to encompass all the vectors of a vector space while the null space has an entirely different set of vectors from the original vector space set.
The Null space in particular doesn't feel like a subset of a matrix's vector space, so I'm likely missing an idea of Vector- and Subspaces. What exactly constitutes a subset of a vector space?
Best Answer
Fir of all we have to be clear about this. There is no such thing as amatrix's vector space. There are a few vector spaces associated to any specific matrix. (And so may vary with matrix to matrix). And then other thing called a vector space consisting of all matrices of a fixed size $m\times n$.
Fix positive integers $m,n$. SO we have two vector spaces $\mathbf{R}^m$ and $\mathbf{R}^n$. Now fix a matrix $A$ of size/shape $m\times n$. Then all linear combinations of columns of $A$ are elements of $\mathbf{R}^m$; so they form a subset closed under the operation of vector addition and scalar multiplication. SUch subsets in any vector space $V$ are called vector subspaces (or simply subspaces) of $V$.
The column space of our $A$ is a vector space obtained as the subspace of $\mathbf{R}^m$ by linear combinations of columns of $A$. Similarly a definition involving rows of $A$ will give the row space, a subspae of $\mathbf{R}^n$.
From the way matrix multiplication is defined it immediately follows that a vector is in the column space of a matrix $A$ iff it is obtainable as the product $Av$ with a vector $v\in \mathbf{R}^n$. Viewed using functions one can see that column space is the range of the following function $T_A\colon \mathbf{R}^n\to \mathbf{R}^m$ defined by $T_A(v) =Av$.
The nullspace of $A$ consists of those vectors $v$ for which $T_A(v)$ is the zero vector.