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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
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

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
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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