4%20Vector%20Spaces

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4 Vector Spaces

• 4.1 Vector Spaces and Subspaces
• 4.2 Null Spaces, Column Spaces, and Linear
  Transformations
• 4.3 Linearly Independent Sets Bases
• 4.4 Coordinate systems

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REVIEW
Definition Let H be a subspace of a vector space
V. An indexed set of vectors
in V is a basis for H if i) is a
linearly independent set, and ii) the subspace
spanned by coincides with H i.e.
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REVIEW

• The Spanning Set Theorem
• Let be a set in V, and
  let .
• If one of the vectors in S, say , is a linear
  combination
• of the remaining vectors in S, then the set
  formed from S by
• removing still spans H.
• b. If , some subset of S is a basis for
  H.

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REVIEW
Theorem The pivot columns of a matrix A form a
basis for Col A.
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4.4 Coordinate Systems
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Why is it useful to specify a basis for a vector
space?

• One reason is that it imposes a coordinate
  system on the vector space.
• In this section well see that if the basis
  contains n vectors, then the coordinate system
  will make the vector space act like Rn.

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Theorem Unique Representation Theorem Suppose
is a basis for V and
is in V. Then For each in V , there exists
a unique set of scalars
such that
.
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Definition Suppose is
a basis for V and is in V. The coordinates
of relative to the basis (the -
coordinates of ) are the weights
such that .

If are the - coordinates
of , then the vector in is the coordinate
vector of relative to , or the -
coordinate vector of .
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Example 1. Consider a basis
for , where Find an x in such that
. 2. For
, find where is the standard basis for
.
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on standard basis
on
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Example For and
, find .
For , let
. Then
is equivalent to .
the change-of-coordinates matrix from to the
standard basis
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The Coordinate Mapping
Theorem Let be a basis
for a vector space V. Then the coordinate
mapping is an one-to-one linear
transformation from V onto .
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Example Let

Determine if x is in H, and if it is, find
the coordinate vector of x relative to .