Linear Algebra: UNIT II: Linear Transformations and Diagonalization
Matrix Representation of a Linear Transformation
Definitions of Matrix Representation of a Linear Transformation
MATRIX REPRESENTATION
OF A LINEAR TRANSFORMATION
Definition 1: Ordered
Basis
Let
V be a finite dimensional vector space. An ordered basis for V is a basis for V
with a specific order.
(ie)
An ordered basis for V is a finite sequence of linearly independent vectors in
V that generates V.
Definition 2:
Let
T:V→W be any linear transformation and let β = { u1, u2,
..... un } and γ= {w1, w2, ... wn}
be the basis for V and W respectively.
Then T (ui) ∈ W.
Any
element in W can be written as a linear combination of γ.
T(ui) = a1iw1 + a2iw2+ ... + amiwn
Definition 3:
The
matrix of order m×n, A defined by Aij=αij, the matrix representation
of T in the ordered bases β and γ and write A = [T]γβ. If
V=W and β=γ, then we write A = [T]γβ.
Note
1.
For the vector space Fn, {e1, e2, e3...
en } is the standard ordered basis for Fn.
2.
For the vector space Pn(F), { 1, x, x2, x3...xn
} is the standard ordered basis for Pn(F).
Definition 4:
Let
β= { u1, u2, u3 ... un } be an
ordered basis for a finite dimensional vector space V. For x ∈ V, let a1, a2, a3... an, be the
unique scalars such that
x = 
aiui
We
define the coordinate vector of x relative to β, denoted by (ie) [x] β = 
Definition 5:
Suppose
that V and W are finite dimensional vector spaces with ordered bases β = { v1,
v2, v3, ... vn } and γ= { w1, w2,
w3 ... wm } respectively. Let T: V→ W be linear. Then for
each j, 1 ≤ j ≤n, there exist unique scalars aij ∈ F, 1 ≤ i ≤ m, such
that T (vj) = mΣi=1 aijwi
for 1 ≤ j ≤ n.
Definition 6:
Let
T, U: V→W be arbitrary functions, where V and W are vector spaces over F, and
let a ∈ F. We define T+U: V→ W
by
(T+U)
(x) = T (x) + U (x) for all x ∈
V, and
aT: V→W by (aT) (x) = aT (x) for x ∈ V.
Definition 7:
Let V and W be vector spaces over F. We
denote the vector space of all linear transformations from V into W by L[V, W].
In the case that V=W, we write L (V) instead of L (V, V).
Linear Algebra: UNIT II: Linear Transformations and Diagonalization : Tag: maths, mathematics : - Matrix Representation of a Linear Transformation
Linear Algebra: UNIT II: Linear Transformations and Diagonalization
Under Subject
Linear Algebra
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