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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?

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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.

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  • $\begingroup$ So for confirmation, subspaces are made of elements of vector spaces. In terms of a linear combination of an m x n matrix, the column space is a subspace of R^(m) made from n vectors that are elements of R^(m). For Null space, by definition, v in Av=0 must be in R^(n) and is thus an element and sub space of R^(n). Right? $\endgroup$ – DexNet34 Nov 5 '18 at 1:43
  • $\begingroup$ Yes. Nullspace of $A$ as above is a subspace of $\mathbf{R}^n$. $\endgroup$ – P Vanchinathan Nov 5 '18 at 2:02
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A matrix doesn't have a vector space as such, a matrix is a representation of a transformation from one vector space to another or itself.

Specifically, if you're talking about a vector space onto itself, we call this an operator, often denoted T, and the matrix will be square. We can then say that the vector space is the sum of the column space of the matrix (which we can also call the range of T) and the null space of the matrix (which we can also call the null of T).

A subset of a vector space is literally just any set of vectors from the vector space (including the infinite set containing all vectors in the space itself). A subspace is a specific type of subset, where those two conditions are held (and they imply the third condition, which is that $\bar{0} \in U$.

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I think your confusion comes from starting with a matrix and looking for "its space".

The space idea comes first. A vector space is a set of objects that behave in a nice way (the axioms for sums and scalar products). So, for example, the set $\mathbb{R}^3$ of triples $(x,y,z)$ of real numbers is a vector space with the usual operations.

Sometimes a subset of a vector space is a vector space in its own right - for example, the set of triples $(x,y,0)$ or the set of triples $(x,x,x)$.

Matrices occur when we want to study particular kinds of functions between vector spaces - so called linear functions. When you have a matrix that represents a linear function the finite set of its columns generates (but is not all of) what we call the column space. It's a subspace of the codomain: the set of vectors there that are actually values of the function. The set of vectors mapped to $0$ is the null space of the transformation. It's a subspace of the domain.

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A vector space $X$ is in the first place an "additive structure" satisfying the rules we associate with such structures, e.g., $a+({-a})=0$, etc. In addition any vector space has associated with it a certain field $F$, the field of scalars for that vector space. The elements $x$, $y\in X$ cannot only be added and subtracted, but they can be as well scaled by "numbers" $\lambda\in F$. The vector $x$ scaled by the factor $\lambda$ is denoted by $\lambda x$. This scaling satisfies the laws we are accustomed to from the scaling of vectors in ${\mathbb R}^3$: $$\lambda(x+y)=\lambda x+\lambda y,\qquad (\lambda+\mu)x=\lambda x+\mu x\ .$$

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