In R2, we are interested in linear transformations f: R2
R2:
| (1) |
f(r x + s y) = r f(x) + s f(y)
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where r and s a (real) scalars and x,y
R2. While working on such
transformations the English mathematician Arthur Cayley (1821-1895) devised in 1857 matrix algebra.
From the definitions of vector addition and componentwise multiplication by a scalar,
| (2) |
x = (x1,x2) = x1(1,0) + x2(0,1)
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Apply now (1) to (2): f(x) = x1f(1,0) + x2f(0,1), where f(1,0) that
should more correctly be written as f((1,0)), is the result of applying f to the vector (1,0). A similar
remark holds for f(0,1) and f(x1,x2) in the following. Let f(1,0)=(f11,f21) and f(0,1)=(f12,f22). Then
| (3) |
f(x) = f(x1, x2) = x1(f11, f21) + x2(f12, f22) = (x1f11 + x2f12, x1f21 + x2f22)
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Assume g is another linear transformation and g(1,0)=(g11, g21) and g(0,1)=(g12, g22). Then by (3),
| (4) |
g(f(x)) = ((x1f11 + x2f12)g11 + (x1f21 + x2f22)g12, (x1f11 + x2f12)g12 + (x1f21 + x2f22)g22)
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or, after regrouping,
| (4') |
g(f(x)) = (x1(f11g11 + f21g12) + x2(f12g11 + f22g12), x1(f11g12 + f21g22) + x2(f12g12 + f22g22))
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We see that a composition g(f(x)) of two linear transformations is in turn linear. Furthermore,
| (5) |
g(f(1,0)) = (f11g11 + f21g12, f11g21 + f21g22), and
|
|
g(f(0,1)) = (f12g11 + f22g12, f12g21 + f22g22)
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Now, a pair of numbers x1,x2 might be written as either a vertical (2x1 matrix) or a horizontal (1x2 matrix) vector.
Given the limitations of HTML, the horizontal convention is a real life saver and has been used so far. However, I must note that the vertical notations are, by far, more common.
Depending on the notations (3) and (5) may be rewritten variously in a vector-matrix format:
| x | (x1, x2) |  |
| F |  |  |
| f(x) | xF | Fx |
| g(f(x)) | xFG | GFx |
What we arrived at is that a linear transformation of a vector space may be expressed as
a product of a matrix and a vector. Composition of two linear transformations is represented
by a product of the corresponding matrices. The claim is more general than what was actually shown.
R2 is known as an arithmetic vector space. The set of all combinations
rsin(x) + scos(x), where x changes over some interval, is another example of a 2-dimensional vector
space whose elements look differently from those of R2. However, as we already remarked,
vector spaces of the same dimensionality are isomorphic, and one way to establish a
correspondence (isomorphism) between them is by selecting bases and identifying their
vectors with tuples of coordinates. For example, a vector rsin(x) + scos(x) could be identified
with an ordered pair (r,s). Under this correspondence, sin(x) and cos(x) appear as (1,0) and (0,1), respectively.