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Examples and matrices
If V and W are finite dimensional and bases have been chosen, then every linear transformation from
V to W can be represented as a matrix; this is useful because it allows concrete calculations. Conversely, matrices yield examples of
linear transformations: if A is a real m-by-n matrix, then the rule f(x) =
Ax describes a linear transformation Rn →
Rm (see Euclidean space).
There are also important examples of linear transformation involving infinite-dimensional spaces. For instance, the integral yields a linear map from the space of all real-valued integrable functions
on some interval to R, while
differentiation is a linear transformation from the space of all differentiable
functions to the space of all functions.
Forming new linear transformations from given ones
The composition of linear transformations is linear: if f : V → W and g :
W → Z are linear, then so is g o f : V → Z.
If f1 : V → W and f2 : V → W
are linear, then so is their sum f1 + f2 (which is defined by (f1 +
f2)(x) = f1(x) + f2(x)).
If f : V → W is linear and a is an element of the ground field K, then
the map af, defined by (af)(x) = a (f(x)), is also linear.
In the finite dimensional case and if bases have been chosen, then the composition of linear maps corresponds to the
multiplication of matrices, the addition of linear maps corresponds ot the addition of
matrices, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.
Endomorphisms and automorphisms
A linear transformation f : V → V is an endomorphism of V; the set of all such endomorphisms End(V) together with addition,
composition and scalar multiplication as defined above forms an associative algebra with identity element over the field K (and in particular a ring). The identity element of this algebra is the identity map id : V → V.
A bijective endomorphism of V is called an automorphism of V. The composition of two automorphisms is again an
automorphism, and the set of all automorphisms of V forms a group, the automorphism group of V
which is denoted by Aut(V) or GL(V).
If V has finite dimension n, then End(V) is isomorphic to the associative algebra of
all n by n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n by n invertible matrices
with entries in K.
Kernel and image
If f : V → W is linear, we define the kernel and the
image of f by
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ker(f) is a subspace of V and
im(f) is a subspace of W. The following dimension formula is
often useful:
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The number dim(im(f)) is also called the rank of f and written as rk(f). If V and
W are finite dimensional, bases have been chosen and f is represented by the matrix A, then the rank
of f is equal to the rank of the matrix A. The
dimension of the kernel is also known as the nullity of the matrix.
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