Download Sections 1.8 and 1.9: Linear Transformations Definitions: 1

Sections 1.8 and 1.9: Linear Transformations
Definitions:
1. Transformation:
A transformation from Rn to Rm is a rule that assigns a vector in Rn to a vector in Rm .
2. Domain:
The domain of a transformation is the input set. For example, if T : Rn → Rm then dom(T ) = Rn .
3. Image:
If x ∈ dom(T ) then T (x) is the image of x under the action of T .
4. Codomain:
The codomain of a transformation is the set containing the images of the domain under T . For example, if
T : Rn → Rm then codom(T ) = Rm .
5. Range:
The range of a transformation is the set of all images
ran(T ) = {T (x) : x ∈ dom(T )}
6. Matrix Transformation:
Let T : Rn → Rm where x 7→ Ax. If T is defined in this way then it is a matrix transformation.
7. Linear Transformation:
A mapping, T , is linear if:
(a) T (u + v) = T (u) + T (v) for all u and v in dom(T ), and
(b) T (cu) = cT (u) for all c ∈ R and for all u ∈ R
8. Linear Transformation (alternate definition):
A mapping, T , is linear if
T (cu + dv) = cT (u) + dT (v) for all u, v ∈ dom(T ) and for all c, d ∈ R.
Note: This definition is the same as the one above. If c = d = 1 we get (a), and if d = 0 we get (b).
Theorems:
Theorem 1. Every matrix transformation is linear.
Proof. Let T (x) = Ax where A is an m × n matrix. Let u, v ∈ dom(T ) = Rn and let c, d ∈ R. To prove that T is
linear we must show that T (cu + dv) = cT (u) + dT (v). Indeed,
T (cu + dv) = A(cu + dv) (definition of T )
= A(cu) + A(dv) (distributing matrix multiplication)
= cAu + dAv
(commuting scalar multiplication)
= cT (u) + dT (v)
(definition of T )
Therefore T is linear.
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Theorem 2. Let T : Rn → Rm be a linear transformation. There there exists a unique matrix A such that
T (x) = Ax
for all x ∈ Rn .
In fact, A is the m × n matrix whose j th column is the vector T (ej ):
h
A = T (e1 )
T (e2 )
···
i
T (en )
Question: Let T : Rn → Rm . What do you need to know in order to generate A?
Proof. (insert proof here)
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Special Transformations: Let T : Rn → Rm be a linear transformation.
1. T is one-to-one if each b ∈ Rm is the image of at most one x ∈ Rn
2. T is onto if each b ∈ Rm is the image of at least one x ∈ Rn
Questions:
1. If T : R3 → R4 is it possible for T to be one-to-one? Explain.
2. If T : R3 → R4 is it possible for T to be onto? Explain.
3. If T : R4 → R3 is it possible for T to be one-to-one? Explain.
4. If T : R4 → R3 is it possible for T to be onto? Explain.
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Theorem 3. Let T : Rn → Rm be a linear transformation. T is one-to-one IF AND ONLY IF the equation T (x) = 0
Theorem 4. Let T : Rn → Rm be a linear transformation such that T (x) = Ax.
(a) T is onto IF AND ONLY IF the columns of A span
(b) T is one-to-one IF AND ONLY IF the columns of A are
Example:
Let T : R4 → R3 such that

x1 + 2x2


T (x) = 2x2 + x4 + x3  .
x2 − x4

1. Find the standard matrix A such that T (x) = Ax.
2. What are the domain, codomain, and range of T ?
3. Is T one-to-one?
4. Is T onto?
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