8/14/2019 translate in swedish (and finnish)-- -- --
--###../translate in swedish (and finnish)-- -- --
--###../translate in s
1/3
Transpose is a function from Rm;n to Rn;m. If A 2 Rm;n; At 2
Rn;m is the matrixwhose (i; j) term is the (j; i) term of A. So row
i (column i) of A becomes columni (row i) of At: If A is an
n-dimensional row vector, then At is an n-dimensional
column vector. If A is a square matrix, At is also
square.Theorem 1) (At)t = A2) (A + B)t = At + Bt3) If c 2 R; (Ac)t
= Atc4) (AB)t = BtAt5) If A 2 Rn, then A is invertible i At is
invertible.In this case (A 1)t = (At) 1.Proof of 5) Suppose A is
invertible. Then I = It = (AA 1)t = (A 1)tAt.Exercise Characterize
those invertible matrices A 2 R2 which have A 1 = At.Show that they
form a subgroup of GL2(R).Triangular Matrices
If A 2 Rn, then A is upper (lower) triangular provided ai;j = 0
for all i > j (allj > i). A is strictly upper (lower)
triangular provided ai;j = 0 for all i j (all j i).A is diagonal if
it is upper and lower triangular, i.e., ai;j = 0 for all i 6= j:
Notethat if A is upper (lower) triangular, then At is lower (upper)
triangular.Theorem If A 2 Rn is strictly upper (or lower)
triangular, then An = 0.Proof The way to understand this is just
multiply it out for n = 2 and n = 3:The geometry of this theorem
will become transparent later in Chapter 5 whenthematrix A denes an
R-module endomorphism on Rn (see page 93).Denition If T is any
ring, an element t 2 T is said to be nilpotent provided 9n
such that tn = 0. In this case, (1 t) is a unit with inverse 1 +
t + t2 + + tn 1.
Transpose is a function from Rm;n to Rn;m. If A 2 Rm;n; At 2
Rn;m is the matrixwhose (i; j) term is the (j; i) term of A. So row
i (column i) of A becomes columni (row i) of At: If A is an
n-dimensional row vector, then At is an n-dimensionalcolumn vector.
If A is a square matrix, At is also square.Theorem 1) (At)t = A2)
(A + B)t = At + Bt3) If c 2 R; (Ac)t = Atc
4) (AB)t = BtAt5) If A 2 Rn, then A is invertible i At is
invertible.In this case (A 1)t = (At) 1.Proof of 5) Suppose A is
invertible. Then I = It = (AA 1)t = (A 1)tAt.Exercise Characterize
those invertible matrices A 2 R2 which have A 1 = At.Show that they
form a subgroup of GL2(R).Triangular MatricesIf A 2 Rn, then A is
upper (lower) triangular provided ai;j = 0 for all i > j (allj
> i). A is strictly upper (lower) triangular provided ai;j = 0
for all i j (all j i).A is diagonal if it is upper and lower
triangular, i.e., ai;j = 0 for all i 6= j: Notethat if A is upper
(lower) triangular, then At is lower (upper) triangular.Theorem If
A 2 Rn is strictly upper (or lower) triangular, then An = 0.Proof
The way to understand this is just multiply it out for n = 2 and n
= 3:The geometry of this theorem will become transparent later in
Chapter 5 when
8/14/2019 translate in swedish (and finnish)-- -- --
--###../translate in swedish (and finnish)-- -- --
--###../translate in s
2/3
thematrix A denes an R-module endomorphism on Rn (see page
93).Denition If T is any ring, an element t 2 T is said to be
nilpotent provided 9nsuch that tn = 0. In this case, (1 t) is a
unit with inverse 1 + t + t2 + + tn 1.
Transpose is a function from Rm;nto Rn;m. If A 2 Rm;n; At 2 Rn;m
is thematrixwhose (i; j) term is the (j; i) term of A.So row i
(column i) of A becomescolumni (row i) of At: If A is an
n-dimensionalrow vector, then At is an n-dimensionalcolumn vector.
If A is a square matrix,At is also square.Theorem 1) (At)t = A2) (A
+ B)t = At + Bt3) If c 2 R; (Ac)t = Atc4) (AB)t = BtAt5) If A 2 Rn,
then A is invertible i At isinvertible.In this case (A 1)t = (At)
1.Proof of 5) Suppose A is invertible.Then I = It = (AA 1)t = (A
1)tAt.Exercise Characterize those invertible
matrices A 2 R2 which have A 1 = At.Show that they form a
subgroup ofGL2(R).Triangular MatricesIf A 2 Rn, then A is upper
(lower)triangular provided ai;j = 0 for all i > j(allj > i).
A is strictly upper (lower)triangular provided ai;j = 0 for all i j
(all ji).A is diagonal if it is upper and lower
triangular, i.e., ai;j = 0 for all i 6= j:Notethat if A is upper
(lower) triangular,then At is lower (upper) triangular.Theorem If A
2 Rn is strictly upper (orlower) triangular, then An = 0.Proof The
way to understand this is justmultiply it out for n = 2 and n =
3:The geometry of this theorem willbecome transparent later in
Chapter 5when thematrix A denes an R-moduleendomorphism on Rn (see
page 93).Denition If T is any ring, an element t 2
T is said to be nilpotent provided 9nsuch that tn = 0. In this
case, (1 t) is aunit with inverse 1 + t + t2 + + tn 1.
Transpose is a function from Rm;n toRn;m. If A 2 Rm;n; At 2 Rn;m
is the matrixwhose (i; j) term is the (j; i) term of A.So row i
(column i) of A becomescolumni (row i) of At: If A is an
n-dimensionalrow vector, then At is an n-dimensionalcolumn vector.
If A is a square matrix,At is also square.Theorem 1) (At)t = A2) (A
+ B)t = At + Bt3) If c 2 R; (Ac)t = Atc4) (AB)t = BtAt5) If A 2 Rn,
then A is invertible i At is
invertible.In this case (A 1)t = (At) 1.Proof of 5) Suppose A is
invertible.Then I = It = (AA 1)t = (A 1)tAt.Exercise Characterize
those invertiblematrices A 2 R2 which have A 1 = At.Show that they
form a subgroup ofGL2(R).Triangular MatricesIf A 2 Rn, then A is
upper (lower)triangular provided ai;j = 0 for all i > j
(allj > i). A is strictly upper (lower)triangular provided
ai;j = 0 for all i j (all ji).A is diagonal if it is upper and
lowertriangular, i.e., ai;j = 0 for all i 6= j:Notethat if A is
upper (lower) triangular,then At is lower (upper)
triangular.Theorem If A 2 Rn is strictly upper (orlower)
triangular, then An = 0.Proof The way to understand this is
justmultiply it out for n = 2 and n = 3:The geometry of this
theorem will
