4.1 Vector Spaces and Subspaces 4.2 Null Spaces, Column Spaces, and Linear Transformations 4.3...
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Transcript of 4.1 Vector Spaces and Subspaces 4.2 Null Spaces, Column Spaces, and Linear Transformations 4.3...
4.1 Vector Spaces and Subspaces4.2 Null Spaces, Column Spaces, and Linear Transformations4.3 Linearly Independent Sets; Bases4.4 Coordinate systems
4 Vector Spaces
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, andii) the subspace spanned by coincides with H; i.e.
pbb ,,1
pbbH ,,Span 1
REVIEW
The Spanning Set Theorem
Let be a set in V, and let .
a. 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.
pvvS ,,1 pvvH ,,Span 1
kv
kv
0H
REVIEW
Theorem
The pivot columns of a matrix A form a basis for Col A.
REVIEW
4.4 Coordinate Systems
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 we’ll see that if the basis contains n vectors, then the coordinate system will make the vector space act like Rn.
Theorem: Unique Representation Theorem
Suppose is a basis for V and is in V. Then there exists a unique set of scalars such that
.
pbb ,,1 pcc ,,1
ppcc bbx 11
x
Definition:
Suppose is a basis for V and is in V. Thecoordinates of relative to the basis (the - coordinates of )are the weights such that .
pbb ,,1
pcc ,,1
If are the - coordinates of , then the vector in
is the coordinate vector of relative to , or the - coordinatevector of .
pcc ,,1
pc
c
x 1
xx x
x nR
xx
ppcc bbx 11
Example:1. Consider a basis for , where
Find an x in such that .
2. For , find where is the standard basis for .
21,bb 2R
1
0,
1
221 bb
2R
3
2x
1
4x x 2R
on standard basis on
1
4x
3
2x
http://webspace.ship.edu/msrenault/ggb/linear_transformations_points.html
The Coordinate Mapping
Theorem
Let be a basis for a vector space V. Then the coordinate mapping is a one-to-one and onto linear transformation from V onto .
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x[ ]βx
€
[ ]
€
b1,L ,bn{ }
][xx nR
1
4x
€
x[ ]β
1
2,
1
121 bb
Example:
For and , find .
For , let .
Then is equivalent to .
€
b1,L ,bp{ }
€
Pβ = b1,L ,bp[ ]
ppbcbcx 11
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x = Pβ x[ ]β
: the change-of-coordinates matrix from β to the standard basis
P
Example:
Let
Determine if x is in H, and if it is, find the coordinate vector of x relative to .
},&,{,
12
3
7
,
0
1
1
,
6
3
2
2121 vvxvv
}.,Span{ 21 vvH
Application to Discrete MathLet = {C(t,0), C(t,1), C(t,2)} be a basis for
P2, so we can write each of the standard basis elements as follows:
C(t,0) = 1(1) + 0t + 0t2
C(t,1) = 0(1) + 1t + 0t2
C(t,2) = 0(1) – ½ t + ½ t2
This means that following matrix converts polynomials in the “combinatorics basis” into polynomials in the standard basis:
€
Mβ =
1 0 0
0 1 −1/2
0 0 1/2
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Application to Discrete MathRecall that = {C(t,0), C(t,1),
C(t,2)} is a basis for P2.
The following matrix converts polynomials in the “combinatorics basis” to polynomials in the standard basis:
Therefore, the following matrix converts polynomials in the the standard basis to polynomials in “combinatorics basis”:
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Mβ−1 =
1 0 0
0 1 1
0 0 2
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
€
Mβ =
1 0 0
0 1 −1/2
0 0 1/2
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
The polynomial p(t) = 3 + 5t – 7t2 has the following coordinate vector in the standard basis S = {1, t, t2}:
S
t
7
5
3
)(p
We now want to find the coordinate vector of p(t) in the “combinatorics basis” = {C(t,0), C(t,1), C(t,2)}:
14
2
3
7
5
3
200
110
001
That is, p(t) = 3 C(t,0) – 2 C(t,1) – 14 C(t,2)
Application to Discrete Math
Application to Discrete MathFind a formula in terms of n for the following
sum
Solution. Using the form we found on the previous slide
€
3+ 5k − 7k 2( )
k =0
n
∑
€
3+ 5k − 7k 2
k =0
n
∑ = 3C(k,0) − 2C(k,1) −14C(k,2)k =0
n
∑
= 3C(n +1,1) − 2C(n +1,2) −14C(n +1,3)
Aside: Why are these true?
http://webspace.ship.edu/deensley/DiscreteMath/flash/ch5/sec5_3/hockey_stick.html
€
C(k,2)k =0
n
∑ = C(n +1,3)€
C(k,0)k =0
n
∑ = C(n +1,1)
€
C(k,1)k =0
n
∑ = C(n +1,2)
Application to Discrete MathFind the matrix M that
converts polynomials in the “combinatorics basis” into polynomials in the standard basis for P3:
Use this matrix to find a version of the following expression in terms of the standard basis:
€
3C(n +1,1) − 2C(n +1,2) −14C(n +1,3)
Application to Discrete MathFind a formula in terms of n for the following
sum
Solution. Continued…..
€
3+ 5k − 7k 2( )
k =0
n
∑
€
3+ 5k − 7k 2
k =0
n
∑ = 3C(k,0) − 2C(k,1) −14C(k,2)k =0
n
∑
= 3C(n +1,1) − 2C(n +1,2) −14C(n +1,3)
= ___(n +1) + ___(n +1)2 + ___(n +1)3
Final Steps…We can finish by multiplying by a matrix
that converts vectors written in {1,(t+1),(t+1)2,(t+1)3} coordinates to a vector written in terms of the standard basis.
Find such a matrix and multiply it by the answer on the previous slide to get a final answer of the form
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___(1) + ___(n) + ___(n2) + ___(n3)