Download Linear Algebra, II

Math 241 Section F1H, Spring 2014
More Linear Algebra: Linear transformations, determinants,
and the conversion factor in the change of variables formula
Recap: Definition and key properties of linear functions
In the following f is assumed to be a function from an m-dimensional space Rm to an n-dimensional space Rn .
• Definition of a linear function: f is called linear if it satisfies (i) f (cx) = cf (x) and (ii) f (x + y) = f (x) + f (y) for
any vectors x, y and any scalar constant c.
• Linear functions are the functions that preserve linear combinations: That is, linear functions are those that
satisfy f (c1 x1 + · · · + ck xk ) = c1 f (x1 ) + · · · + ck f (xk ) for any constants c1 , . . . , ck and vectors x1 , . . . , xk .
• Linear functions correspond to matrices. The linear functions from Rm to Rn are exactly the functions of the
form f (x) = Ax, where A is an n × m matrix. The matrix A associated with a linear function f is the derivative
matrix of that function: A = Df .
• A linear function is uniquely determined by its values on the basis vectors. For example, if f : R3 → R3 is
a linear function, knowing the three values a = f (i), b = f (j), c = f (k), one can compute f at an arbitrary vector in
R3 : f (c1 , c2 , c3 ) = f (c1 i + c2 j + c3 k) = c1 a + c2 b + c3 c.
• The columns of the matrix A associated with a linear function are the images of the basis vectors.
h 2 4 For
i
1
example, a linear function that maps i to h2, 2, 4i, j to h4, 8, 1i and k to h1, 5, 5i has an associated matrix A = 2 8 5 .
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Geometric properties of linear functions from Rn to Rn
In the following we focus on linear function that map the n-dimensional space Rn to itself, and the n × n matrices associated
to such linear functions. These functions are also called linear transformations.
• The n-dimensional unit cube is mapped to an n-dimensional parallelepiped (i.e., an “n-dimensional parallelogram”),
spanned by the images of the basis vectors, i.e., the column vectors of the associated matrix A. For example, if
A = [ 31 11 ], then the unit square is mapped to the parallelogram spanned by the vectors Ai = h3, 1i and Aj = h1, 1i.
• The standard unit grid is mapped to the lattice spanned by the images of the basis vectors, i.e., the column vectors of
the matrix A.
• Volumes of regions are “blown up” by a constant factor, regardless of the location of the region.
• The “blow-up factor” is the volume of the parallelepiped spanned by the images of the basis vectors, i.e., the column
vectors of the associated matrix A. For example, if A = [ 31 11 ], the “blow-up factor” is the area of the parallelogram
spanned by h3, 1i and h1, 1i.
• The volume of this parallelepiped, and hence the blow-up factor, is the absolute value of the determinant of the
associated matrix. For example, if A = [ 31 11 ], then det A = 3 · 1 − 1 · 1 = 2, so the blow-up factor is | det A| = 2; if
A = [ 13 11 ], then det A = 1 · 1 − 3 · 1 = −2, so the blow-up factor is | det A| = | − 2| = 2.
Determinants: Definition and algebraic properties
In the following all matrices are assumed to be square matrices of the same dimension n. The determinant, det A, of an
n × n matrix A has the following properties.
• Recursive definition and computation:
• The determinant of a 1 × 1 matrix A = [a] is simply the matrix entry a.
• The determinant of a 2 × 2 matrix A = ac db is defined as det A = ad − bc.
• Determinants of higher-dimensional matrices are defined recursively as follows: If A is an n × n matrix with first
row a11 , . . . , a1n , then det A = a11 det A11 − a12 det A12 + a13 det A13 − + . . . , where A1i is the (n − 1) × (n − 1)
matrix obtained by deleting the 1st row and i-th column from A.
• Determinants of triangular and diagonal matrices:
triangular matrix is a matrix with all 0’s in the triangle
" λ1 A
0 0 0 ... #
above the main diagonal, i.e., a matrix of the form A =
∗ λ2 0 0 ...
∗ ∗ λ3 0 ...
...
...
∗ ∗ ... ∗ λn
, where the asterisks (∗) denote arbitrary entries.
The determinant of a triangular matrix is the product of the diagonal entries: det A = λ1 λ2 . . . λn . In particular, this
formula applies to a diagonal matrix with entries λ1 , . . . , λn on the diagonal, and 0 else.
• Algebraic properties:
• det I = 1. (Here I is the identity matrix, the matrix whose entries are 1 on the main diagonal, and 0 else.)
• det AT = det A. (Here AT is the transpose of the matrix A, obtained by interchanging rows with columns; i.e.,
if the column vectors of A are a1 , . . . , an , then AT is the matrix whose rows are given by a1 , . . . , an , written as
row vectors.)
• det(AB) = det A det B. (Here AB is the usual matrix multiplication.)
• det A−1 = (det A)−1 if det A 6= 0. (Here A−1 is the inverse matrix to A, defined by AA−1 = I. Any matrix with
det A 6= 0 has a unique inverse matrix.)
Determinants: Geometric meaning
For our context the fundamental property of determinants is its connection to volumes given as follows:
• The determinant of A is, up to ± sign,1 equal to the volume of the parallelepiped spanned by the column
vectors of A.
• The absolute value of the determinant of A is the “blow-up factor” of the linear transformation associated with the matrix matrix A, i.e., the factor by which the transformation given by A “blows up”
volumes.
Explanation of the conversion factor in the change of variables formula
• Setup: Assume we are given a change of variables x = T(u), where x = hx1 , . . . , xn i is the standard rectangular
coordinate system in Rn , u = hu1 , . . . , un i denotes another coordinate system in Rn , and T is the conversion function
between the u and x coordinates. We seek to derive the “correct” conversion factor needed to convert an integral with
respect to dx1 . . . dxn to an integral with respect to du1 . . . dun .
• Special case: T linear. If the conversion function T is linear, then the above analysis applies directly and shows
that T “blows up” volumes by a constant factor | det A|, where A is the matrix associated to T. Thus, | det A| is the
“correct” conversion factor in the case of a linear transformation of variables.
Example: In R2 , the change of variables (x, y) = (au, bv) = T(u, v), where a, b are nonzero constants, is a linear
function with associated matrix A = [ a0 0b ], so the conversion factor is | det A| = |ab|.
• General case. When T is not linear, we approximate T by linear functions. From the theory of derivatives in Rn we
know that the best-matching linear approximation to T at a given point u is the linear function whose
associated matrix is DT(u), the derivative matrix of T at the point u. Applying the above analysis to this
linear approximation, we obtain | det DT(u)| as the appropriate conversion factor at a given point u.
• Jacobian determinant and ∂-notation: If T(u1 , . . . , un ) = (x1 , . . . , xn ), then the Jacobian determinant
is defined as the determinant
∂x1
∂x1 . . . ∂u
n
∂(x1 , . . . , xn ) ∂u1
= det DT
= ...
∂(u1 , . . . , un ) ∂xn
∂xn . . . ∂un
∂u1
∂(x1 ,...,xn )
∂(u1 ,...,un )
Hence the “correct” conversion factor derived above, | det DT|, is just the absolute value of the Jacobian of the trans∂(x1 ,...,xn )
formation involved, i.e., | ∂(u
|.
1 ,...,un )
Example: For the polar coordinate transformation T(r, θ) = (r cos θ, r sin θ), we have det DT(r, θ) = ∂(x,y)
∂(r,θ) =
cos θ −r sin θ = r cos2 θ + r sin2 θ = r, giving the familiar conversion factor r for polar coordinates. Note that, in
sin θ r cos θ
contrast to the above example, in this case the conversion factor is not constant, but depends one the polar variables.
1 The precise determination of the ± sign is a bit tricky, and you don’t need to be concerned with this here. For applications to multiple integrals,
the sign is irrelevant since the conversion factor is defined as the absolute value of the determinant of the transformation, so any minus sign gets
killed.