Row and column picture
One of the most important insights of linear algebra is obtained from
the role of columns and rows. The columns are helpful to describe the
image, the rows helpful to describe the kernel.
The column picture tells that
A x = x_{1} v_{1} + ... + x_{n} v_{n}
is a linear combination of columns.
The row picture tells that A x = 0 means that
x is perpendicular to the rows of A. One can state this as
ker(A)^{T} = im (A^{T})
which relates the transposed matrix A^{T} of the matrix A
with the orthogonal complement V^{T} of the column space V.
Data
Duality reflects also in statistics
If you have two data sets like x = (1,3,1,2,3) and y = (2,1,5,5,1).
Then there are two ways to describe these data. The first is to see
the data as two vectors in 5 dimensional space. The cosine of the
angle between X = xE[x] and Y = yE[y] is called the correlation
of the data. The second dual picture is to visualize the data
points (1,2),(3,1),(1,5),(2,5),(3,1) in the plane.
The regression line passes through the point E[x],E[y]
and has the slope m = cos(alpha) Y/X = (X.Y)/X^{2}
because Ym X is minimized if m X is the projection of Y onto X
Again, if the data are written as a matrix, then the two pictures
correspond to taking the transpose of the matrix.
A =  1 2 
 3 1 
 1 5 
 2 5 
 3 1 

A^{T} =
 1 3 1 2 3 
 2 1 5 5 1 

