BASIS

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BASIS
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A vector space containing infinitely many vectors
can be efficiently described by listing a set of
vectors that SPAN the space.
eg describe the solutions to
reduces to
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A vector space containing infinitely many vectors
can be efficiently described by listing a set of
vectors that SPAN the space.
eg describe the solutions to
reduces to
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A vector space containing infinitely many vectors
can be efficiently described by listing a set of
vectors that SPAN the space.
eg describe the solutions to
reduces to
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A vector space containing infinitely many vectors
can be efficiently described by listing a set of
vectors that SPAN the space.
eg describe the solutions to
reduces to
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A vector space containing infinitely many vectors
can be efficiently described by listing a set of
vectors that SPAN the space.
eg describe the solutions to
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A vector space containing infinitely many vectors
can be efficiently described by listing a set of
vectors that SPAN the space.
eg describe the solutions to
These two vectors SPAN the set of solutions. Each
of the infinitely many solutions is a linear
combination of these two vectors!
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A spanning set can be an efficient way to
describe a vector space containing infinitely
many vectors.
SPANS R2 - but is it the most efficient way to
describe R2 ?
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A spanning set can be an efficient way to
describe a vector space containing infinitely
many vectors.
SPANS R2 - but is it the most efficient way to
describe R2 ?
Why do we need this? It is a linear combination
of (depends on) the other two.
This independent set still spans R2 , and is a
more efficient way to describe the vector space!
definition An INDEPENDENT set of vectors
that SPANS a vector space V is called a BASIS
for V.
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?
? SPANS R2
Given any x and y there exist c1 and c2 such that
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?
? is INDEPENDENT
A linear combination of these vectors
produces the zero vector ONLY IF c1 and c2 are
both zero.
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?
? is INDEPENDENT and ? SPANS R2 .
Therefore ? is a
BASIS for R2.
is called the standard basis for R2
? is a nonstandard basis - why do we need
nonstandard bases?
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Consider the points on the ellipse below
Described relative to the ? basis they
are solutions to 9x2 4y2 1
Described relative to the standard basis they
are solutions to 8x2 4xy 5y2 1
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example
not a BASIS for R2
?
v
There are lots of different ways to write v as a
linear combination of the vectors in the set ?
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theorem If ?
is a BASIS for a vector space V,
then for every vector
in V there are unique scalars

Such that
the cs exist because ? spans V
they are unique because ? is independent
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theorem If ?
is a BASIS for a vector space V,
then for every vector
in V there are unique scalars

Such that
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ONLY IF
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theorem If ?
is a BASIS for a vector space V,
then for every vector
in V there are unique scalars

Such that
the coordinates of
relative to the ? basis
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Coordinates
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This is the vector v
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Relative to the standard basis the coordinates
of v are
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?
1
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Relative to the ? basis the coordinates of v are

3
2
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(No Transcript)
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v

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Theorems about bases
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example Suppose V is a vector space that is
SPANNED by the two vectors
Is it possible that this set of three vectors is
INDEPENDENT ?
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example Suppose V is a vector space that is
SPANNED by the two vectors
Is it possible that this set of three vectors is
INDEPENDENT ?
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example Suppose V is a vector space that is
SPANNED by the two vectors
Is it possible that this set of three vectors is
INDEPENDENT ?
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example Suppose V is a vector space that is
SPANNED by the two vectors
Is it possible that this set of three vectors is
INDEPENDENT ?
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example Suppose V is a vector space that is
SPANNED by the two vectors
Is it possible that this set of three vectors is
INDEPENDENT ?
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example Suppose V is a vector space that is
SPANNED by the two vectors
Is it possible that this set of three vectors is
INDEPENDENT ?
IF

IF
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example Suppose V is a vector space that is
SPANNED by the two vectors
Is it possible that this set of three vectors is
INDEPENDENT ?
NO
IF
3 VECTORS CAN NEVER BE INDEPENDENT in a VECTOR
SPACE that is SPANNED BY 2 VECTORS
The rank is less than the number of variables
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The number of independent vectors in a vector
space V can never exceed the number of vectors
that span V.
theorem
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Two different bases for the same vector space
will contain the same number of vectors.
is a basis for V
proof
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Two different bases for the same vector space
will contain the same number of vectors.
If
is a basis for V
theorem
then k m
is a basis for V
and
definition
The number of vectors in a basis for V is called
the DIMENSION of V.
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An independent set of vectors that does not span
V can be padded to make a basis for V.
theorem
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A spanning set that is not independent can be
weeded to make a basis.
theorem
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theorem
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theorem
If the dimension of V is n then the set
n
is INDEPENDENT IF AND ONLY IF it SPANS V
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theorem
If the dimension of V is n then the set
n
is INDEPENDENT IF AND ONLY IF it SPANS V
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theorem
If the dimension of V is n then the set
n
is INDEPENDENT IF AND ONLY IF it SPANS V
If S
If S
spans V
is independent then S spans V.
then S is independent.
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theorem
If the dimension of V is n then the set
n
is INDEPENDENT IF AND ONLY IF it SPANS V
If S
If S
spans V
is independent then S spans V.
then S is independent.
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theorem
If the dimension of V is n then the set
n
is INDEPENDENT IF AND ONLY IF it SPANS V