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Lecture 7Hyper-planes, Matrices, and
Linear SystemsShang-Hua Teng
Line in 2D
• By linear equation
12 yx
12 yx x=3y=1
Line in 2D
• By a point and a vector: passing (3,1) along vector (2,1)
ty
tx
ttyx
1
23
:)1,2()1,3(),(
set edParametriz
12
have we, geliminatinBy
yx
t
x=3y=1
Line in 2D
• By two points: passing (3,1) and (0,-1/2)
ttty
x
tty
x
:5.0
0
1
3)1(
have we term,same theCombine
:1
3
5.0
0
1
3
set edParametriz
12
have we, geliminatinBy
yx
t
(3,1)
(0,-1/2)
Line and Affine Combination in 2D
• The line passing two points or the affine combination of two points is given by
tb
at
b
at
b
a
b
a
b
a
b
a
:)1(
,affine,line
points twoofn Combinatio Affineor Line
2
2
1
1
2
2
1
1
2
2
1
1
System of Linear Equations (2D)
• Row Picture[conventional view]: two lines meets at a point
1123
12
yx
yx
1123 yx
12 yx x=3y=1
System of Linear Equations (2D)
• Column Picture: linear combination of the first two vectors produces the third vector
1123
12
yx
yx
11
1
2
2
3
1yx
And geometrically• Column Picture: linear combination of the
first two vector produce the third vector
11
1
2
2
3
1yx
2
2
11
1
3
1
3
13
x=3y=1
Coefficient Matrix and Matrix-Vector Product
1123
12
yx
yx
11
1
2
2
3
1yx
11
1
23
21
y
x
A 2 by 2 matrix is a square table of 4 numbers, two per row and two per column
System of Linear Equations (3D)
• Row Picture[conventional view]: Three planes meet at a single point
236
4252
632
zyx
zyx
zyx
2
0
0
• Row Picture[conventional view]: Two planes meet at a single line A line and a plane meet at a single point
Intersection of Planes
System of Linear Equations (3D)
• Column Picture: linear combination of the first three vectors produces the fourth vector
2
4
6
1
2
3
3
5
2
6
2
1
zyx
2
0
0
236
4252
632
zyx
zyx
zyx
Coefficient Matrix and Matrix-Vector Product
2
4
6
136
252
321
z
y
x
A 3 by 3 matrix is a square table of 9 numbers, three per row and three per column
236
4252
632
zyx
zyx
zyx
2
4
6
1
2
3
3
5
2
6
2
1
zyx
Matrix Vector Product (by row)
• If A is a 3 by 3 matrix and x is a 3 by 1 vector, then in the row picture
xrow
xrow
xrow
Ax
)2 (
)2 (
)1 (
Matrix Vector Product (by column)
• If A is a 3 by 3 matrix and x is a 3 by 1 vector, then in the row picture
3
2
1
321
where
)3 ()2 ()1 (
x
x
x
x
columnxcolumnxcolumnxAx
More about 3D Geometry
• Points and distance, Balls and Spheres– 0 dimension in 3 dimensions
• Lines– 1 dimension in 3 dimensions
• Plane– 2 dimensions in 3 dimensions
Line in 3D
• 2D– By linear equation – A point and a vector– Two points
• Affine combination
• 3D– A point and a vector– Two points
• Affine combination
Line in 3D
• By a point and a vector: passing p along vector v
zz
yy
xx
tvpz
tvpy
tvpx
ttvp
:
Line and Affine Combination in 3D• The line passing two points or the affine combination of
two points is given by
t
v
vv
t
u
uu
t
v
vv
u
uu
v
vv
u
uu
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
:)1(
,affine,line
points twoofn Combinatio Affineor Line
Plane in 3D
• Line in 2D– By linear equation – Affine combination of two points
• “Every” two points determine a line
• 3D– By linear equation– Affine combination of three points
• “Every” three points determine a plane
Linear Equation and its Normal
3
2
1
so
032
implying
632
632
have, we,, and ,, :on points any twofor
632:),,(
21
21
21
212121
222
111
222111
zz
yy
xx
zzyyxx
zyx
zyx
zyxzyxP
zyxzyxP
Normal of a Plane
Plane and Affine Combination in 3D
)1(
,,affine
,,plane
321
321
321
ppp
ppp
ppp
1p
2p
3p
u
v
s
tsst
psptsstppssuv
pttpu
)1(,
)1()1( )1(
)1(
3213
21
High Dimensional Geometric Extension• Points and distance, Balls and Spheres
– 0 dimension in n dimensions
• Lines– 1 dimension in n dimensions
• Plane– 2 dimensions in n dimensions
• k-flat – k-dimensions in n dimensions
• Hyper-plane – (n-1)-dimensions in n dimensions
Affine Combination in n-D
1
,,,affine
1
12211
21
k
k
j j
k
ppp
ppp
Hyper-Planes in d-D
• Line in 2D– By linear equation – Affine combination of two points
• 3D– By linear equation– Affine combination of three points
• n-D– By linear equation– Affine combination of n-1 points
Linear Equation and its Normal
n
n
k kkk
n
k kk
n
k kk
nn
n
k kkn
aaaa
where
ayx
yxa
bya
bxa
yyyxxxP
bxaxxxP
,,,
)(
so
0)(
implying
, ,,,y and ,,, x:on points Any two
:),,,(
21
1
1
1
2121
121
Matrix (Uniform Representation for Any Dimension)
• An m by n matrix is a rectangular table of mn numbers
ji
nmmm
n
n
ajiA
aaa
aaa
aaa
A
,
,2,1,
,22,21,2
,12,11,1
),( write weSometime
...
...
...
...
