Page 170 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 15: Tensors in Cartesian Coordinates

Thus, formula (4) defines the vectorial object y. As a result, we have vector y determined by a Notes
linear operator F and by vector x. Therefore, we write
y = F(x) ...(7)
and say that y is obtained by applying linear operator F to vector x. Some people like to write (7)
without parentheses:
y = Fx. ....(8)
Formula (8) is a more algebraic form of formula (7). Here the action of operator F upon vector
x is designated like a kind of multiplication. There is also a matrix representation of formula (8),
in which x and y are represented as columns:

y 1 F 1 1 F 2 1 F 3 1 x 1
2
2
y  F 1 2 F 2 2 F 3 2 x . ...(9)
y 3 F 1 3 F 2 3 F 3 3 x 3
Exercise 15.6: Derive (9) from (4).
Exercise 15.7: Let  be some real number and let x and y be two vectors. Prove the following
properties of a linear operator (7):
(1) F(x + y) = F(x) + F(y),
(2) F( x) = F(x).
Write these equalities in the more algebraic style introduced by (8). Are they really similar to
the properties of multiplication?
Exercise 15.8: Remember that for the product of two matrices
det(AB) = detA detB. ...(10)
Also remember the formula for det(A ). Apply these two formulas to (3) and derive
–1
det F = detF. ...(11)

Formula (10) means that despite the fact that in various bases linear operator F is represented by
various matrices, the determinants of all these matrices are equal to each other. Then we can
define the determinant of linear operator F as the number equal to the determinant of its matrix
in any one arbitrarily chosen basis e , e , e :
1
3
2
det F = det F. ...(12)
Exercise 15.9 (for deep thinking). Square matrices have various attributes: eigenvalues,
eigenvectors, a characteristic polynomial, a rank (maybe you remember some others). If we
study these attributes for the matrix of a linear operator, which of them can be raised one level
up and considered as basis-independent attributes of the linear operator itself? Determinant (12)
is an example of such attribute.

Exercise 15.10: Substitute the unit matrix for F into (1) and verify that F is also a unit matrix in
i 
i
j
j
this case. Interpret this fact.
Exercise 15.11: Let x = e for some basis e , e , e in the space. Substitute this vector x into (7) and
i
1
2
3
by means of (4) derive the following formula:
3
j
F(e ) =  F e . ...(13)
j
i
i
j 1

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