for x=(xy xn) € R", define the li -ToRm by ||selle₁ = ½ |xil lacil || i=1 Rove that is actually a norm, namely t satisfies the three axioms of a horm and plot the unit ball in R², namely the set points xER² such that lll 1 2 x₂1 (0,1) (-0-5,0-5) Q 1₂0) --05,0.5) 2 (05105) (1,03² 24 •(0-5,-05) unit ball 1) 11x111₁ =1 Q₁ ទ ((0-1)
for x=(xy xn) € R", define the li -ToRm by ||selle₁ = ½ |xil lacil || i=1 Rove that is actually a norm, namely t satisfies the three axioms of a horm and plot the unit ball in R², namely the set points xER² such that lll 1 2 x₂1 (0,1) (-0-5,0-5) Q 1₂0) --05,0.5) 2 (05105) (1,03² 24 •(0-5,-05) unit ball 1) 11x111₁ =1 Q₁ ទ ((0-1)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Hello, I would like some guidance with this question. To prove that the norm satisfies the 3 properties. Is it a case where I would have to choose a set of numbers that fall on the unit ball and put them into the L1 norm equation
Transcribed Image Text:ER
For x=(x₁, x₂)² =R", define the Is
hoRm by || selle₁= 2/2 |acil
i=1
PRove that is actually a horm, namely
it satisfies the three axioms of a horm,
and plot the unit ball in R²,
; namely
the set points xER² such that llell |
x21 (0,1)
(-0-5,0-5)
-1,₂0)
(05/05)
2
(0-8 10-5)
294
(1,0)
•(0.5,-05) unit ball
21(8-1)
Dllxkld
||(20²₁1x₂)|1=1 = |x₂|+|2₁|=1
디
Q₁
X₂=1= 31
Q₂ = x₂ = 1 + 2
Q₁ ==x₂-2²₁=1 = x ₂ = -2₁-1
Q₁ = = x₂ + x1=1 =) X₂=X₁-1
Phove 3 properties:
x = (x₁₂x₂)
11x11₁ = 1x₁1+1x₂1
D||xc||20 iff x=0
Sincellx11=1
Then 1x₁1 + 1x21=1
from the unit ball you can see
the set of points agree with this
That is 11x417/0
|||+|0l = 1
10051+10.51=1
10-251+/0²251=1
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