AA Denitions of Determinants 47
111 112 (113
provided the determinant r121 r122 (.123 of the coefcient matrix is nonzero.
13] 15132 5133
When we calculate these solutions explicitly, this determinant
2.4 The Kernel and the Range of L 63
Example Let L : R3 :- R3 be given by its action on the standard basis in R3
1 1 0 1 0 3
0 7 0 6 l 2
x x
To nd the image L y of any y E R3, we expand this vector in
L!
2.5 The Quotient Space V /ker L and the Isomorphism 1lv*,.,_/l(er E ran L 65
the subspace LlN(cfw_1,2, . . . ,k, 17k+1 , it, . . . ,) will be equal to V, since every
subspace of V which has the dim
52 A Determinants
all at?! am
11 12 I b11b12'b1n
' b21b22'b2n
B = cfw_It11 Eli12 tilt1a = . . :
t+11 +12 i+ln ' ' ' '
bnl bnz bun
:11 HQ art
so that det B : (1)1 det A (the row-interchange property).
2.6 Representation Theory 71
The isomorphism v induced by the choice of the basis 12 in V is the bijection
which associates with every .1? E V, which must be rst expanded in this basis
i = i1=1 xiio,
AZ Properties of Determinants 53
the row-exchange property, we have det B : det B, or det B = O. The cofactor
expansion of det B, i.e.,
n
det B : Z (1)+kbfk det Bfk = 0
k=1
implies, due to 3,1, : AH,
64 2 Linear Mappings and Linear Systems
We also write concisely ran L : L(V,).
To prove that the above sets are indeed subspaces, we shall show that they are
both closed when one takes any linear co
2.6 Representation Theory 69
An arbitraryL: V.1 :- Wm is given if we know the 1mages cfw_L(v1) L(v2). L(vn)
of the basis vectors cfw_v1.v2, . . . ,vn from V. in W. . We can uniquely expand these
image
Chapter 2
Linear Mappings and Linear Systems
2.1 A Short Plan for the First 5 Sections of Chapter 2
Since the whole concept of modern mathematics is based on sets and their mappings,
we shall now, aft
55 A Determinants
that the minor matrices Ag,- and ij are equal for each j = 1,2,. . .n. Therefore,
(-1)i+jiAiji=(-1)i+j|3cli
and computing det B by the Laplace expansion along the i-th row, we have
d
54 A Determinants
Proof Let us consider a lower triangular n >< n matrix A and perform in suc-
cession the Laplace expansion along the rst row in its determinant and in the
resulting cofactors:
(1110
68 2 Linear Mappings and Linear Systems
The linmap L : V :~ Wm itself is then represented by an m X a real matrix M,
which is also the linmap M : R > R3. Matrix algebra offers many methods for solv-
i
2.2 Some General Statements about Mapping 61
(Note the difference in notation: l for elements instead of z- for sets)
The set of all images is a subset of B, and it is called the range of f (ran f):
r
65 2 Linear Mappings and Linear Systems
Suppose that some 331 E l/ is also a preimage of 372 : L(.:E]) : j) and prove that .f]
must be from a? -| ker L:
1261):? and L(.f) :j:
=>L(f1)L(x)=w=>L(f1x)=w=>
2.6 Representation Theory 6?
(Notice that we replaced the symbol Vn/NL for the inverse images of L in V by
Vn/ker L since inverse images are now the equivalence classes of the equivalence
relation whi
45 A Determinants
Here, we note the signicance of the scalar
151115712 H251
. . r1 1) .
for the coefclent matrlx A = [a] b1 ] . To nd a more compact form for thls scalar,
2 2
. . .
43 A Determinants
It is called the cfw_apiece expansion by the rst row of the original determinant. The
three smaller determinants are called cofactors, and they are the signed determinants
of 2 X 2 s
A2 Properties of Detenninants 49
(the three positive products are calculated by following the multiplications in the
rst scheme, and the negative ones according to the second scheme).
Example
61 8
311
A2 Properties of Determinants 51
Indeed, we perform the Laplace expansion of det A along the rst row
11
det A = 2(1)]+ka]kdetil1k.
k2]
where A is an n X n matrix. For det B = det AT, we perform the co
62 2 Linear Mappings and Linear Systems
2.3 The Denition of Linear Mappings (Linmaps)
Let V and Wm be two real vector spaces of dimensions at and m, respectively. A
linear mapping (a iinmop) from V to
70 2 Linear Mappings and Linear Systems
111.X1+I12.X2+'+t11nxn J31
a21x1+223rZ2+-+12nxn JZ
: l-I-I III! III! III! : . : WI:
ail-x] +m2x2 + ' +amnxn ym
or. in short. [aij]m><rt[xi]nxl = [ydmxls where
60 2 Linear Mappings and Linear Systems
columns from R and R, respectively, and L itself by an m X a matrix M . Making
use of matrix algebra, we can now easily solve these problems for L (nding ker L
A2 Properties of Detenninants
Example Calculate the deteninant of the 4 X 4 matrix:
2 1 4 3
1 1 0 2 we add column 2 to column 1
3 2 3 1 and (2) column 3 to column 4,
1 2 2 3
1 1 4 1
_ 0 1 0 0 and perf
A.2 Properties of Determinants 55
Now suppose that A is an invertible matrix. We can use the row-reduction method
analogous to the G] method (see Sect. 2.13), but without making the leading
entries eq