1.9 The Matrix of a Linear Transformation

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• 1.9 The Matrix of a Linear Transformation

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Example Suppose T is a linear transformation
from to such that
and . With no
additional information, find a formula for the
image of an arbitrary x in .
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Theorem 10. Let T a linear
transformation. Then there exists a unique
matrix A such that for a x in
. In fact, A is the matrix whose
jth column is the vector where is the
jth column of the identity matrix in .

Note A is called the standard matrix for the
linear transformation T.
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Example Find the standard matrix A for the
dilation transformation T(x)4x, where
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Find the standard matrix of each of the following
transformations.
Reflection through the x-axis
Reflection through the y-axis
Reflection through the yx
Reflection through the y-x
Reflection through the origin
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Find the standard matrix of each of the following
transformations.
Horizontal Contraction Expansion
Vertical Contraction Expansion
Projection onto the x-axis
Projection onto the y-axis
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Definition A mapping is said
to be onto if each b in is the image
of at least one x in .
Definition A mapping is said
to be one-to-one if each b in is the
image of at most one x in .
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Theorem Let be a linear
transformation. Then, T is one-to-one if and
only if has only the trivial
solution.
Theorem Let be a linear
transformation with the standard matrix A. 1. T
is onto iff the columns of A span . 2. T is
one-to-one iff the columns of A are linearly
independent