Introduction%20to%20Linear%20Transformations

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• Section 1.8
• Introduction to Linear Transformations

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Recall that the difference between the matrix
equation
and the associated vector equation
is notation.
However, the matrix equation can
arise is linear algebra (and applications) in a
way that is not directly connected with linear
combinations of vectors. This happens when we
think of a matrix A as an object that acts on a
vector by multiplication to produce a new
vector
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Example
A

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Recall that is only defined if the
number of columns of A equals the number of
elements in .
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A
So multiplication by A transforms into
.
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• In the previous example, solving the equation
  can be thought of as finding all
  vectors in that are transformed into
  the vector in under the action
  of multiplication by A.

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Transformation
Function or Mapping
T
T
Range
Domain
Codomain
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Let A be an mxn matrix.
Matrix Transformation
Codomain
A
Domain
b
x
A
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Example The transformation T is defined by
T(x)Ax where T .
For each of the following determine m and n.
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Matrix Transformation
Axb
x
A
b
Domain
Codomain
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Linear Transformation
Definition A transformation T is linear if (i)
T(uv)T(u)T(v) for all u, v in the domain of
T (ii) T(cu)cT(u) for all u and all scalars c.
Theorem If T is a linear transformation,
then T(0)0 and T(cudv)cT(u)dT(v) for all u,
v and all scalars c, d.