Change the matrix by multiplying one column by a number. Consider a positive definite matrix A. We can think of the matrix as a linear transformation. Now supopse we get matrix B by multiplying only one column of A by a number. Is there a geometric relation between A and B? 
 A: 
Is there a geometric relation between $A$ and $B$?

Yes, because all matrices that are linear transformations can be decomposed into the following operations: stretching (scaling), rotation, and reflection. 
So, the geometric relationship between matrix $A$ and $B$ is simply that $B$ is scales/rotates/reflects slightly differently than $A$. 
To know exactly how it differs, one might have to consider some sort of matrix factorization, such as SVD, which can decompose a matrix into pure scaling/rotation/reflection matrices.
A: To give a concrete algebraic formulation to this, suppose you want to multiply the $j$th column by $\lambda$.  Let $L$ be the matrix 
$$
L \;\; =\;\; \left [ \begin{array}{ccccc}
1 &&&& \\
& \ddots &&& \\
&& \underbrace{\lambda}_{jth \; column} && \\
&&& \ddots & \\
&&&& 1 \\
\end{array} \right ]
$$
with zeros on all non-diagonal entries.  Then $B = AL$.
A: We have that all Matrix $A\in M_{n\times m}(\mathbb{R})$, $m\leq n$ has a vectorial subspace of $\mathbb{R}^n$ asociated. if $V$ is the vectorial subspace generated for the columns of $A$. Then we can see that each element of $V$ has the form: 
$$v=Ax, x\in \mathbb{R}^m.$$ 
Now if $A=(a_{1},\dots, a_{m})$, where $a_{i}$ are columns of $A$, we can write $B=(a_{1},\dots,\lambda a_{i},\dots, a_{m})$. Then: 
$$Bx=\sum_{j\neq i}^{m}x_{j}a_{j}+x_{i}\lambda a_{i}.$$ Where $x_{j}$ are the components of vector $x$. Then we can to see this as a dilation or contraction of the component $v_ {i}$ of the vector $v$ in the direction of column $a_ {i}$.
