Cartesian coordinate space
René Descarte (1596-1650)
French philosopher, mathematician, physicist, and writer.
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n-dimensional Cartesian coordinate space Rn
∼= R× . . .× R (n factors)Rn is the totality of all ordered n-tuples (x1, . . . , xn) wherexi ∈ Rfor n = 2 is the (x , y) ∈ R2
πi : Rn → R defined by
πi((x1, . . . , xn)) = xi
is called the i th coordinate function or i th coordinateprojectionGiven a function f : A→ Rn,define fi := πi ◦ fthese fi : A→ R completely determines f
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Algebraic structure of Rn
For x = (x1, . . . , xn),y = (y1, . . . , yn) define
x + y = (x1 + y1, . . . , xn + yn)
Note: usual laws of addition,
0 = (0, . . . ,0),−x = (−x1, . . . ,−xn)
Scalar multiplication: αx := (αx1, . . . , αxn)
1. associative: α(βx) = (αβ)x2. distributive: α(x + y) = αx + αy3. identity: 1x = x
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Linear maps
Definitionf : Rn → Rm is said to be a linear map iff (αx + βy) = αf (x) + βf (y).
Examples: Projection map πi , multiplication by scalardot product by a fixed vector; what about the converse?f : Rn → Rm is linear iff fi s are linearDistance travelled is a linear function of time when velocityis constant. So is the voltage as a function of resistancewhen the current is constant. The logarithm of the changein concentration in any first order chemical reaction is alinear function of time.|x |, xn (n > 1), sin x , etc. are not linear
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Exercise:
(i) Show that if f is a linear map then
f (k∑
i=1
αixi) =k∑
i=1
αi f (xi).
(ii) Show that the projection on a line L passing through theorigin defines a linear map of R2 to R2 and its image isequal to L.
(iii) Show that rotation through a fixed angle θ is a linear mapfrom R2 → R2.
(iv) By a rigid motion of Rn we mean a map f : Rn → Rn suchthat
d(f (x), f (y)) = d(x,y),
where d(x,y) =√∑n
i=1(xi − yi)2, x,y ∈ Rn
Show that a rigid motion of R3 which fixes the origin is alinear map.
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Structure of linear maps
L(n,m) = Set of all linear maps from Rn to Rm
For f ,g ∈ L(n,m) define αf and f + g by
(αf )(x) = αf (x); (f + g)(x) = f (x) + g(x)
If f ∈ L(n,m) and g ∈ L(m, l), then g ◦ f ∈ L(n, l)If f ,g ∈ L(n,1), then define fg : Rn → R by
(fg)(x) = f (x)g(x).
Does fg ∈ L(n,1)?Let ei = (0, . . . ,0,1,0, . . . ,0) (standard basis elements). Ifx ∈ Rn, then x =
∑ni=1 xiei .
If f ∈ L(n,m), then
f (x) =∑
i
xi f (ei)
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if given v1, . . . , vn ∈ Rm, define a (unique) linear map f byassigning f (ei) = vi
Examples:1. Given a f ∈ L(n,1), if we put u = (f (e1), . . . , f (en)), then
f (x) =∑
i xi f (ei) = u.x2.
a11x1 + a12x2 + . . . + a1nxn = b1
a21x1 + a22x2 + . . . + a2nxn = b2
. . . . . .
am1x1 + am2x2 + . . . + amnxn = bm
Set of all solutions of j th equation is a hyper plane Pj in Rn.Solving the system means finding
P1 ∩ . . . ∩ Pm.
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On the other hand the lhs of each of these equations can bethought of as a linear map Ti : Rn → R. Together, they defineone function
T ∈ L(Rn,Rm)
such that T = (T1, . . . ,Tm).
Determining x ∈ Rn such that T (x) = b,where b = (b1, . . . ,bm)
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Matrix representation
(x1, x2, . . . , xn) is a row vectorx1x2...
xn
= (x1, x2, . . . , xn)T ( where T stands for
transpose) is called a column vector and elements of Rn
are denoted by this.Given linear map f : Rn → Rm we get n column vectors (ofsize m) viz., f (e1), . . . , f (en). Place them side by side:For instance, if f (ej) = (f1j , f2j , . . . , fmj)
T , then we obtain
Mf =
f11 f12 . . . f1nf21 f22 . . . f2n...
...fm1 fm2 . . . fmn
This array is called a matrix with m rows and n columns.
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We say matrixMf is of size m × n.Notation: Mf = ((fij))Matrices are equal if their sizes are the same and theentries are the same.If m = 1 we get row matrices; if n = 1 we get columnmatrices
M : L(n,m)→ Mm,nf 7→ Mf
is one-one and called matrix representation of linear maps
Mf+g =Mf +Mg ; Mαf = αMf
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Examples:
1. MId = I or In where
I =
1 0 . . . 00 1 . . . 0...
...0 0 . . . 1
= ((δij))
δij = 1 if i = j and = 0 otherwise(Kronecker delta)
2. Linear T : R2 → R2 which interchange coordinates is
represented by(
0 11 0
)3. Corresponding to multiplication by α ∈ R is the diagonal
matrix D(α, . . . , α) = ((αδij))
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