58
Dot Product and Length
iT
h
The dot productof two vectors X =
x1 x2 · · ·
xn
iT
h
and Y
y1 y2 · · ·
yn
in
Rn is defined by
X.Y = X T Y = x1 y1 + x2 y2 + · · · + xn yn .
iT
h
is defined by
The length of the vector x1 x2 · · · xn
kXk =
√
q
X.X =
x21 + x22 + · · · + x2n .
Properties of dot product are :
1. X.Y = Y.X
2. X.(Y + Z) = X.Y + X.Z
3. X.(aY ) = (aX).Y = a(X.Y ) for any number a.
4. kXk2 = X.X
5. kXk ≥ 0 and kXk = 0 if and only if X = 0.
6. kaXk = |a|kXk for scalars a.
7. If X and Y are vectors in Rn , then |X.Y | ≤ kXkkY k. (Cauchy Inequality)
8. kX + Y k ≤ kXk + kY k. (Triangle Inequality)
Definition 17. A vector X ∈ Rn is said to be unit vector if kXk = 1. Two vectors X and
Y in Rn are said to be orthogonal if X.Y = 0. A non-zero set of vectors {X1 , X2 , · · · , Xk }
in Rn is called an orthogonal set of vectors if
Xi .Xj = 0, for all i 6= j.
If in addition kXi k = 1 for all i, then {X1 , X2 , · · · , Xk } is called orthonormal set.
Example 63. The standard basis of Rn is an orthonormal set.
59
Theorem 36. (Pythagoras’ Theorem) If {X1 , X2 , · · · , Xk } is an orthonormal set of vectors
in Rn , then
kX1 + X2 + · · · + Xk k2 = kX1 k2 + kX2 k2 + · · · + kXk k2 .
Theorem 37. Gram-Schmidt Algorithm. If {X1 , X2 , · · · , Xk } is any basis for a subspace
of Rn then {F1 , F2 , · · · , Fk } is an orthogonal basis for U where
F1 = X1
X2 .F1
F1
kF1 k2
X3 .F1
X3 .F2
= X3 −
F1 −
F2
2
kF1 k
kF2 k2
..
.
Xk .F1
Xk .F2
Xk .Fk−1
= Xk −
F1 −
F2 − · · · −
Fk−1
2
2
kF1 k
kF2 k
kFk−1 k2
F2 = X2 −
F3
Fk
60
Linear Transformations
A map T : R2 → R2 is called a linear transformation if
1. T (X + Y ) = T (X) + T (Y ) for all X and Y in R2 .
2. T (rX) = rT (X) for all X ∈ R2 and r ∈ R.
Example 64. Let A be a 2 × 2 matrix. Define,
T : R2 → R2
by T (X) = AX. This map is called the matrix transformation induced by A. We can check
that it is a linear transformation.
Example 65. The counterclockwise rotation about origin ·through a¸ right angle is a matrix
0 −1
transformation and the corresponding matrix is equal to
.
1 0
Theorem 38. A linear transformation
T : R2 → R2
is a matrix transformation and the corresponding matrix is
and E2 are columns of the identity.
£
¤
T (E1 ) T (E2 ) , where E1
·
2
2
1
2
¸
Example 66. Let T : R → R be a linear transformation. If T
=
· ¸ · ¸
1
5
T
=
, find the corresponding matrix of this linear transformation.
0
4
Solution:
· ¸ · ¸
· ¸
1
1
0
=
+2
= E1 + 2E2
2
0
1
· ¸
·
¸
1
3
T
= T (E1 + 2E2 ) = T (E1 ) + 2T (E2 ) =
2
−1
· ¸
· ¸ · ¸
·
¸
5
3
5
−1
and T (E1 ) =
, therefore T (E2 ) = 12 (
−
) = −5 .
4
1
4
2
·
¸
£
¤
5 −1
Thus, A = T (E1 ) T (E2 ) =
.
4 −5
2
·
3
−1
¸
and
Example 67. The identity map IR2 : R2 → R2 is defined by IR2 X = X for all X ∈ R2 .
Then, the corresponding matrix for this transformation is identity.
61
Definition 18. If T : R2 → R2 and S : R2 → R2 are two linear transformation, their
composition
T ◦ S : R2 → R2
is defined by T ◦ S(X) = T (S(X)) for all X ∈ R2 . A linear transformation T : R2 → R2
is called invertible if there exists a linear transformation S : R2 → R2 such that S ◦ T =
T ◦ S = IR2 .
Theorem 39. If T : R2 → R2 and S : R2 → R2 are two linear transformation with
corresponding matrices A and B respectively, the corresponding matrix of their composition
T ◦ S : R2 → R2
is equal to BA.
Theorem 40. Let T : R2 → R2 be a linear transformation with the corresponding matrix
A. Then, T is invertible if and only if A is invertible.