Linear Algebra

Topic: Geometry of Linear Maps 275
And neither is this transformation of R3 , which projects vectors into the xzplane.
� �
x
y
z
↦→
−→
� �
x
0
z
But even in higher-dimensional spaces, the situation isn’t complicated. Of
course, any linear map h: R n → R m can be represented with respect to, say,
the standard bases by a matrix H. Recall that any matrix H can be factored as
H = PBQ where P and Q are nonsingular and B is a partial-identity matrix.
And, recall that nonsingular matrices factor into elementary matrices, matrices
that are obtained from the identity matrix with one Gaussian step
I kρi
−→ Mi(k) I ρi↔ρj
−→ Pi,j
I kρi+ρj
−→ Ci,j(k)
(i �= j, k �= 0). Thus we have the factorization H = TnTn−1 ...TjBTj−1 ...T1
where the T ’s are elementary. Geometrically, a partial-identity matrix acts as a
projection, as here. (That is, the map that this matrix represents with respect
to the standard bases is a projection. We say that this is the map induced by
the matrix.)
⎛ ⎞
x
⎝y⎠
z
�
1 0
�
0
0 1 0
0 0 0
−→
E3 , E3 ⎛ ⎞
x
⎝y⎠
0
Therefore, we will have a description of the geometric action of h if we just
describe the geometric actions of the three kinds of elementary matrices. The
pictures below sticks to the elementary transformations of R 2 only, for ease of
drawing.
The action of a matrix of the form Mi(k) (that is, the action of the transformation
of R 2 that is induced by this matrix) is to stretch vectors by a factor
of k along the i-th axis. This is a dilation. This map stretches by a factor of 3
along the x-axis.
� �
x
y
↦→
−→
� �
3x
y
Note that if 0 ≤ k