Linear Algebra: UNIT II: Linear Transformations and Diagonalization
Matrix Representation of a Linear Transformation: Theorems
Important Theorems for Engineering Maths or Mathematics - Matrix Representation of a Linear Transformation: Theorems
MATRIX REPRESENTATION
OF A LINEAR TRANSFORMATION - THEOREMS
Theorem 7
Let V and W be finite
dimensional vector spaces with ordered bases β and γ respectively, and let T,
U: V→W be linear transformations.
Then (a) [T+U]γβ = [T]γβ + [U]γβ
and
(b) [aT]γβ
= a[T]γβ for all scalars a.
Proof:
Let
β ={v1, v2, v3 ... vn} and γ= { w1,
w2, w3 ... wm }. There exist unique scalars aij
and bij (1 ≤ i ≤ m, 1 ≤ j
≤n) such that
Proof
of (b) is similar.
Theorem 8
Let V and W be vector
spaces over a field F, and let T,U: V→W be linear.
(a) For all a ∈ F, aT + U is linear.
(b) Using the
operations of addition and scalar multiplication, the collection of all linear
transformations from V to W is a vector space over F.
Proof:
(a)
Let x, y = V and c ∈
F.
Then
(aT+U) (cx+y)= aT (cx + y) + U (cx+y)
=
a [T (cx+y)] + cU (x) + U (y)
=
a [cT (x) + T (y)] + cU (x) + U (y)
=
acT (x)+cU (x) + aT (y) + U (y)
=
c(aT+U) (x) + (aT + U) (y)
So
aT + U is linear.
(b)
Note that T0, the zero transformations, play the role of the zero
vector, it is easy to verify that the axioms of a vector space are satisfied
and hence that the collection of all linear transformations from V into W is a
vector space over F.
Linear Algebra: UNIT II: Linear Transformations and Diagonalization : Tag: maths, mathematics : - Matrix Representation of a Linear Transformation: Theorems
Linear Algebra: UNIT II: Linear Transformations and Diagonalization
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Linear Algebra
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