econ 205 - slides from lecture 6 - university of california...
TRANSCRIPT
![Page 1: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/1.jpg)
Econ 205 - Slides from Lecture 6
Joel Sobel
August 30, 2010
Econ 205 Sobel
![Page 2: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/2.jpg)
Linear Algebra Basics
Definitionn-dimensional Euclidean Space:
Rn = R× R× · · · × R× R
If X and Y are sets, then
X × Y ≡ {(x , y) | x ∈ X , y ∈ Y }
soRn = {(x1, x2, . . . , xn) | xi ∈ R,∀ i = 1, 2, . . . , n}
Econ 205 Sobel
![Page 3: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/3.jpg)
Interpretations:
I Calculus: x ∈ Rn as a list of n real numbers; x = (x1, . . . , xn).
I Linear algebra x ∈ Rn is column vector. When we do this wewrite
x =
x1x2...xn
.
I Sometimes an element in Rn is best thought of as a direction.
Econ 205 Sobel
![Page 4: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/4.jpg)
DefinitionThe zero element:
0 =
00...0
Definition (Vector Addition)
For x, y ∈ Rn we have
x + y =
x1 + y1x2 + y2
...xn + yn
Vector addition is commutative
x + y = y + xEcon 205 Sobel
![Page 5: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/5.jpg)
Definition (Scalar Multiplication)
For x ∈ Rn, and a ∈ R we have
ax =
ax1ax2
...axn
In other words every element of x gets multiplied by a.
Econ 205 Sobel
![Page 6: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/6.jpg)
Vector Spaces
Definition (Vector Space)
A vector space is a set V in which the operation of addition andmultiplication by a scalar make sense, and in which operations arecommutative and associative. In addition:0 + v = v (additive identity)for each v there is a −v (additive inverse)
A subset of a vector space that is itself a vector space is called asubspace.Euclidean Spaces (Rn) are the leading example of vector spaces.We will need to talk about subsets of Euclidean Spaces that have alinear structure (they contain 0, and if x and y are in the set, thenso is x + y and all scalar multiples of x and y).
Econ 205 Sobel
![Page 7: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/7.jpg)
Matrices
DefinitionAn m × n matrix is an element of Mm×n written as in the form
A =
α11 α12 · · · α1n
α21 α22 · · · α2n...
......
αm1 αm2 · · · αmn
= [αij ]
where m denotes the number of rows and n denotes the number ofcolumns.
Note An m × n matrix is just of a collection of nm numbersorganized in a particular way. Hence we can think of a matrix asan element of Rm×n. The extra notation Mm×n makes it possibleto distinguish the way that the numbers are organized.
Econ 205 Sobel
![Page 8: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/8.jpg)
Note Vectors are just a special case of matrices. e.g.
x =
x1x2...xn
∈Mn×1
This notation emphasizes that we think of a vector with ncomponents as a matrix with n rows and 1 column.
Econ 205 Sobel
![Page 9: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/9.jpg)
A2×3
=
(0 1 56 0 2
)
Econ 205 Sobel
![Page 10: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/10.jpg)
DefinitionThe transpose of a matrix A, is denoted At . To get the transposeof a matrix, we let the first row of the original matrix become thefirst column of the new (transposed) matrix.
At =
α11 α21 · · · α1n
α12 α22 · · · α2n...
......
α1m α2m · · · αnm
= [αji ]
DefinitionA matrix A is symmetric if A = At .
So we can see that if A ∈Mm×n, then At ∈Mn×m.
Econ 205 Sobel
![Page 11: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/11.jpg)
Continuing the example, we see that
At
3×2=
0 61 05 2
Econ 205 Sobel
![Page 12: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/12.jpg)
Matrix Algebra
[Addition of Matrices] If
Am×n
=
α11 α12 · · · α1n
α21 α22 · · · α2n...
......
αm1 αm2 · · · αmn
= [αij ]
and
Bm×n
=
β11 β12 · · · β1nβ21 β22 · · · β2n
......
...βm1 βn2 · · · βmn
= [βij ]
then
Econ 205 Sobel
![Page 13: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/13.jpg)
A + Bm×n
= Dm×n
=
α11 + β11 α12 + β12 · · · α1n + β1nα21 + β21 α22 + β22 · · · α2n + β2n
......
...αm1 + βm1 αm2 + βm2 · · · αmn + βmn
= [δij ] = [αij+βij ]
A + B︸ ︷︷ ︸m×n
=
α11 + β11 α12 + β12 · · · α1n + β1nα21 + β21 α22 + β22 · · · α2n + β2n
......
...αm1 + βm1 αm2 + βm2 · · · αmn + βmn
= [αij+βij ]
Econ 205 Sobel
![Page 14: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/14.jpg)
Definition (Multiplication of Matrices)
If Am×k
and Bk×n
are given, then we define
Am×k· Bk×n
= Cm×n
= [cij ]
such that
cij ≡k∑
l=1
ailblj
so note above that the only index being summed over is l.
Econ 205 Sobel
![Page 15: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/15.jpg)
Let
A2×3
=
(0 1 56 0 2
)and
B3×2
=
0 31 02 3
Then
A2×3· B3×2︸ ︷︷ ︸
2×2
=
(0 1 56 0 2
)·
0 31 02 3
=
((0× 0) + (1× 1) + (5× 2), (0× 3) + (1× 0) + (5× 3)(6× 0) + (0× 1) + (2× 2), (6× 3) + (0× 0) + (2× 3)
)=
(11 154 24
)
Econ 205 Sobel
![Page 16: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/16.jpg)
In general:A · B 6= B · A
For example
A2×3· B3×46= B
3×4· A2×3
(The product on the right is not defined.)
DefinitionAny matrix which has the same number of rows as columns isknown as a square matrix, and is denoted A
n×n.
Econ 205 Sobel
![Page 17: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/17.jpg)
DefinitionThere is a special square matrix known as the identity matrix. Anymatrix multiplied by this identity matrix gives back the originalmatrix. The Identity matrix is denoted In and is equal to
Inn×n
==
1 0 . . . 00 1 . . . 0...
. . ....
0 . . . 0 1
.
Econ 205 Sobel
![Page 18: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/18.jpg)
DefinitionA square matrix is called a diagonal matrix if aij = 0 wheneveri 6= j .
DefinitionA square matrix is called an upper triangular matrix (resp. lowertriangular if aij = 0 whenever i > j (resp. i < j).
Diagonal matrices are easy to deal with. Triangular matrixes arealso somewhat tractable. You’ll see that for many applications youcan replace an arbitrary square matrix with a related diagonalmatrix.
Econ 205 Sobel
![Page 19: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/19.jpg)
For any matrix Am×n
we have the results that
Am×n· In = A
m×n
andIm · A
m×n= A
m×n
Econ 205 Sobel
![Page 20: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/20.jpg)
DefinitionWe say a matrix A
n×nis invertible or non-singular if ∃ B
n×nsuch that
An×n· Bn×n︸ ︷︷ ︸
n×n
= Bn×n· An×n︸ ︷︷ ︸
n×n
= In
If A is invertible, we denote it’s inverse as A−1.So we get
A(n×n)
· A−1
(n×n)︸ ︷︷ ︸n×n
= A−1
n×n· An×n︸ ︷︷ ︸
n×n
= In
A square matrix that is not invertible is called singular.
Econ 205 Sobel
![Page 21: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/21.jpg)
DefinitionThe determinant of a matrix A (written detA = |A|) is definedinductively.
n = 1 A(1×1)
detA = |A| ≡ a11
n ≥ 2 A(n×n)
detA =| A |≡
a11|A−11| − a12|A−12|+ a13|A−13| − · · · ± a1n|A−1n|
where A−1j is the matrix formed by deleting the firstrow and jth column of A.Note A−1j is an (n− 1)× (n− 1) dimensional matrix.
Econ 205 Sobel
![Page 22: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/22.jpg)
Examples
If
A2×2
= [aij ] =
(a11 a12a21 a22
)=⇒ detA = a11a22 − a12a21
Econ 205 Sobel
![Page 23: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/23.jpg)
If
A3×3
= [aij ] =
a11 a12 a13a21 a22 a23a31 a32 a33
=⇒ detA = a11
∣∣∣∣ a22 a23a32 a33
∣∣∣∣− a12
∣∣∣∣ a21 a23a31 a33
∣∣∣∣+ a13
∣∣∣∣ a21 a22a31 a32
∣∣∣∣
Econ 205 Sobel
![Page 24: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/24.jpg)
Determinant
The determinant is useful primarily because of the following result:
TheoremA matrix is invertible if and only if its determinant 6= 0.
Econ 205 Sobel
![Page 25: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/25.jpg)
DefinitionThe adjoint of a matrix A
n×n(adj (A)) is the n × n matrix with
entry ij equal toadj A = (−1)i+j detA−ij
where adj A is the adjoint of A.
DefinitionThe Inverse of a matrix A
n×nis defined as
A−1 =1
detA· adj A
where adj A is the adjoint of A.
Econ 205 Sobel
![Page 26: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/26.jpg)
If A is a (2× 2) matrix and invertible then
A−1 =1
detA·(
a22 −a12−a21 a11
)
Econ 205 Sobel
![Page 27: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/27.jpg)
Inner Product
Definition (Inner Product)
If x, y ∈Mn×1, then the inner product (or dot product or scalarproduct) is given by
xty = x1y1 + x2y2 + · · ·+ xnyn
=n∑
i=1
xiyi
Note that xty = ytx. We will have reason to use this conceptwhen we do calculus, and will write x · y =
∑ni=1 xiyi .
Econ 205 Sobel
![Page 28: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/28.jpg)
Distance
Definition (Euclidean Distance)
d(x, y) = ||x− y||
where
||z|| =√z21 + z22 + · · ·+ z2n
=
√√√√ n∑i=1
z2i
Under the Euclidean metric, the distance between two points is thelength of the line segment connecting the points. We call ||z||,which is the distance between 0 and z the norm of z.Notice that ||z||2 = z · z.
Econ 205 Sobel
![Page 29: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/29.jpg)
Inner Product Properties
When x · y = 0 we say that x and y are orthogonal/at rightangles/perpendicular.Two vectors are perpendicular if and only if their inner product iszero. This fact follows rather easily from “The Law of Cosines.”The law of cosines states that if a triangle has sides A,B, and Cand the angle θ opposite the side c , then
c2 = a2 + b2 − 2ab cos(θ),
where a, b, and c are the lengths of A, B, and C respectively. Thismeans that:
(x− y) · (x− y) = x · x + y · y− 2||x||||x|| cos(θ),
where θ is the angle between x and y.
Econ 205 Sobel
![Page 30: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/30.jpg)
If you multiply everything out you get the identity:
||x||||y|| cos(θ) = xty. (1)
1. The inner product of two non-zero vectors is zero if and onlyif the cosine of the angle between them is zero.
2. Since the absolute value of the cosine is less than or equal toone,
||x||||y|| ≥| xty | .
Econ 205 Sobel
![Page 31: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/31.jpg)
Systems of Linear Equations
Consider the system of m equations in n variables:
y1 = α11x1 + α12x2 + · · ·+ α1nxn
y2 = α21x1 + α22x2 + · · ·+ α2nxn...
yi = αi1x1 + αi2x2 + · · ·+ αinxn...
ym = αm1x1 + αm2x2 + · · ·+ αmnxn
Here the variables are the xj .
Econ 205 Sobel
![Page 32: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/32.jpg)
This can be written as
y(m×1)
= A(m×n)
· x(n×1)︸ ︷︷ ︸
(m×1)
where
y(m×1)
=
y1y2...ym
, x(n×1)
=
x1x2...xn
Am×n
=
α11 α12 · · · α1n
α21 α22 · · · α2n...
......
αm1 αm2 · · · αmn
= [αij ]
or, putting it all together
Econ 205 Sobel
![Page 33: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/33.jpg)
y(m×1)
= A(m×n)
· x(n×1)︸ ︷︷ ︸
(m×1)y1y2...ym
=
α11 α12 · · · α1n
α21 α22 · · · α2n...
......
αm1 αm2 · · · αmn
·
x1x2...xn
Econ 205 Sobel
![Page 34: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/34.jpg)
Facts about solutions to linear equations
DefinitionA system of equations of the form Ax = 0 is called a homogeneoussystem of equations.
A homogeneous system always has a solution (x = 0). The solutionwill not be unique if there are more unknowns than equations or ifthere are as many equations as unknowns and A is singular.
TheoremWhen A is square, the system Ax = y has a unique solution if andonly if A is nonsingular.
If defined, x = A−1y.
If not, then there is a nonzero z such that Az = 0. This meansthat if you can find one solution to Ax = y you can find infinitelymany.
Econ 205 Sobel
![Page 35: Econ 205 - Slides from Lecture 6 - University of California ...econweb.ucsd.edu/~jsobel/205f10/notes6.pdfI Linear algebra x 2Rn is column vector. When we do this we write x = 0 B B](https://reader033.vdocuments.net/reader033/viewer/2022060908/60a342d38521c45ce533de5b/html5/thumbnails/35.jpg)
General Theory
Using “elementary row operations” reduce general system ofequations to canonical form that permits you to identify solutionset easily.
Econ 205 Sobel