The document discusses inner products and their properties in linear algebra. It provides examples of dot products of vectors in R3 that satisfy the properties of an inner product and those that do not. Specifically, it shows that u1w1 + u2w2 + u3w3 and u1w1 + 2u2w2 + 3u3w3 satisfy the properties, while u1w1 + u2w2 - u3w3 does not always satisfy non-negativity. It also discusses other properties such as symmetry, homogeneity, additivity and positive definiteness.

1. Tutorial MTH 3201
Linear Algebras

2. Tutorial 4

3. IMPORTANT!
Let 𝑢 = 𝑢1 , 𝑢2 , 𝑢3 and 𝑣 = 𝑣1 , 𝑣2 , 𝑣3 are vectors in 𝑅 𝑛
𝑢, 𝑣 = 𝑢 ∙ 𝑣 = 𝑢1 𝑣1 + 𝑢2 𝑣2 + 𝑢3 𝑣3
Euclidean inner product on 𝑅 𝑛
*Please develop your writing skills in mathematics

4. Let 𝑢 = 𝑢1 , 𝑢2 , 𝑢3 and 𝑤 = 𝑤1 , 𝑤2 , 𝑤3
Since 𝑢, 𝑤 = 𝑢2 𝑤2 + 𝑢3 𝑤3
𝑘𝑢, 𝑤 = 𝑘𝑢2 𝑤2 + 𝑘𝑢3 𝑤3
𝑢, 𝑤 = 𝑢 ∙ 𝑤 = 𝑢2 𝑤2 + 𝑢3 𝑤3
= 𝑤2 𝑢2 + 𝑤3 𝑢3 = 𝑘𝑤2 𝑢2 + 𝑘𝑤3 𝑢3
= 𝑤∙ 𝑢 = 𝑘 𝑤2 𝑢2 + 𝑤3 𝑢3
= 𝑤, 𝑢 = 𝑘 𝑤, 𝑢
𝑢, 𝑤 = 𝑤, 𝑢 Symmetry Homogeneity 𝑘𝑢, 𝑤 = 𝑘 𝑤, 𝑢
Axiom
𝑢 + 𝑤, 𝑣 = 𝑢, 𝑣 + 𝑤, 𝑣 Additivity Positivity 𝑤, 𝑤 ≥ 0 𝑎𝑛𝑑
𝑤, 𝑤 = 0 𝑖𝑓𝑓 𝑤 = 0
Since 𝑢, 𝑤 = 𝑢2 𝑤2 + 𝑢3 𝑤3
𝑤, 𝑤 = 𝑤 ∙ 𝑤 = 𝑤2 𝑤2 + 𝑤3 𝑤3
𝑢 + 𝑤, 𝑣 = 𝑢2 + 𝑤2 𝑣2 + 𝑢3 + 𝑤3 𝑣3
= 𝑢2 𝑣2 + 𝑤2 𝑣2 + 𝑢3 𝑣3 + 𝑤3 𝑣3 𝑤, 𝑤 = 0 𝑖𝑓𝑓 𝑤2 = 𝑤3 = 0
= 𝑢2 𝑣2 + 𝑢3 𝑣3 + 𝑤2 𝑣2 + 𝑤3 𝑣3 𝐵𝑢𝑡, 𝑤1 𝑐𝑎𝑛 𝑏𝑒 𝑎𝑛𝑦 𝑣𝑎𝑙𝑢𝑒, ≠ 0
= 𝑢∙ 𝑣 + 𝑤∙ 𝑣
= 𝑢, 𝑣 + 𝑤, 𝑣
∴ 𝑢, 𝑤 = 𝑢2 𝑤2 + 𝑢3 𝑤3 are not inner product on 𝑅3

5. Let 𝑢 = 𝑢1 , 𝑢2 , 𝑢3 and 𝑤 = 𝑤1 , 𝑤2 , 𝑤3
Since 𝑢, 𝑤 = 𝑢1 𝑤1 + 2𝑢2 𝑤2 + 3𝑢3 𝑤3
𝑘𝑢, 𝑤 = 𝑘𝑢1 𝑤1 + 2𝑘𝑢2 𝑤2 + 3𝑘𝑢3 𝑤3
𝑢, 𝑤 = 𝑢 ∙ 𝑤 = 𝑢1 𝑤1 + 2𝑢2 𝑤2 + 3𝑢3 𝑤3
= 𝑤1 𝑢1 + 2𝑤2 𝑢2 + 3𝑤3 𝑢3 = 𝑘𝑤1 𝑢1 + 2𝑘𝑤2 𝑢2 + 3𝑘𝑤3 𝑢3
= 𝑤∙ 𝑢 = 𝑘 𝑤1 𝑢1 + 2𝑤2 𝑢2 + 3𝑤3 𝑢3
= 𝑤, 𝑢 = 𝑘 𝑤, 𝑢
𝑢, 𝑤 = 𝑤, 𝑢 Symmetry Homogeneity 𝑘𝑢, 𝑤 = 𝑘 𝑤, 𝑢
Axiom
𝑢 + 𝑤, 𝑣 = 𝑢, 𝑣 + 𝑤, 𝑣 Additivity Positivity 𝑤, 𝑤 ≥ 0 𝑎𝑛𝑑
Since 𝑢, 𝑤 = 𝑢1 𝑤1 + 2𝑢2 𝑤2 + 3𝑢3 𝑤3 𝑤, 𝑤 = 0 𝑖𝑓𝑓 𝑤 = 0
𝑢 + 𝑤, 𝑣 = 𝑢1 + 𝑤1 𝑣1 𝑤, 𝑤 = 𝑤 ∙ 𝑤 = 𝑤1 𝑤1 + 2𝑤2 𝑤2 + 3𝑤3 𝑤3
+2 𝑢2 + 𝑤2 𝑣2 + 3 𝑢3 + 𝑤3 𝑣3
= 𝑢1 𝑣1 + 𝑤1 𝑣1 + 2𝑢2 𝑣2 + 2𝑤2 𝑣2 𝑤, 𝑤 = 0 𝑖𝑓𝑓𝑤1 = 𝑤2 = 𝑤3 = 0
+3𝑢3 𝑣3 + 3𝑤3 𝑣3
= 𝑢1 𝑣1 + 2𝑢2 𝑣2 + 3𝑢3 𝑣3
+ 𝑤1 𝑣1 + 2𝑤2 𝑣2 + 3𝑤3 𝑣3
= 𝑢∙ 𝑣 + 𝑤∙ 𝑣
= 𝑢, 𝑣 + 𝑤, 𝑣
∴ 𝑢, 𝑤 = 𝑢1 𝑤1 + 2𝑢2 𝑤2 + 3𝑢3 𝑤3 are inner product on 𝑅3

6. Let 𝑢 = 𝑢1 , 𝑢2 , 𝑢3 and 𝑤 = 𝑤1 , 𝑤2 , 𝑤3
Since 𝑢, 𝑤 = 𝑢1 𝑤1 + 𝑢2 𝑤2 − 𝑢3 𝑤3
𝑘𝑢, 𝑤 = 𝑘𝑢1 𝑤1 + 𝑘𝑢2 𝑤2 − 𝑘𝑢3 𝑤3
𝑢, 𝑤 = 𝑢 ∙ 𝑤 = 𝑢1 𝑤1 + 𝑢2 𝑤2 − 𝑢3 𝑤3
= 𝑘𝑤1 𝑢1 + 𝑘𝑤2 𝑢2 − 𝑘𝑤3 𝑢3
= 𝑤1 𝑢1 + 𝑤2 𝑢2 − 𝑤3 𝑢3
= 𝑤∙ 𝑢 = 𝑘 𝑤1 𝑢1 + 𝑤2 𝑢2 − 𝑤3 𝑢3
= 𝑤, 𝑢 = 𝑘 𝑤, 𝑢
𝑢, 𝑤 = 𝑤, 𝑢 Symmetry Homogeneity 𝑘𝑢, 𝑤 = 𝑘 𝑤, 𝑢
Axiom
𝑢 + 𝑤, 𝑣 = 𝑢, 𝑣 + 𝑤, 𝑣 Additivity Positivity 𝑤, 𝑤 ≥ 0 𝑎𝑛𝑑
𝑤, 𝑤 = 0 𝑖𝑓𝑓 𝑤 = 0
Since 𝑢, 𝑤 = 𝑢1 𝑤1 + 𝑢2 𝑤2 − 𝑢3 𝑤3
𝑤, 𝑤 = 𝑤 ∙ 𝑤 = 𝑤1 𝑤1 + 𝑤2 𝑤2 − 𝑤3 𝑤3
𝑢 + 𝑤, 𝑣 = 𝑢1 + 𝑤1 𝑣1
+ 𝑢2 + 𝑤2 𝑣2 − 𝑢3 + 𝑤3 𝑣3 𝑤, 𝑤 ≥ 0,
𝑤1 2 + 𝑤2 2 − 𝑤3 2 ≥ 0
= 𝑢1 𝑣1 + 𝑤1 𝑣1 + 𝑢2 𝑣2 + 𝑤2 𝑣2 − 𝑢3 𝑣3 − 𝑤3 𝑣3
= 𝑢1 𝑣1 + 𝑢2 𝑣2 − 𝑢3 𝑣3 + 𝑤1 𝑣1 + 𝑤2 𝑣2 − 𝑤3 𝑣3 𝑤1 2 + 𝑤2 2 ≥ 𝑤3 2
𝑤1 2 + 𝑤2 2 𝑠ℎ𝑜𝑢𝑙𝑑 𝑏𝑒 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑤3 2
= 𝑢∙ 𝑣 + 𝑤∙ 𝑣
𝑤1 2 + 𝑤2 2 < 𝑤3 2
= 𝑢, 𝑣 + 𝑤, 𝑣
∴ 𝑢, 𝑤 = 𝑤1 𝑤1 + 𝑤2 𝑤2 − 𝑤3 𝑤3 are not inner product on 𝑅3

7. 1/2 = 𝑤1 2 + 𝑤2 2
𝑤 = 𝑤, 𝑤 = 22 + (−5)2 = 29
𝑤 = 5(2) 2 + 2(−5)2 = 70
−2 3 2 −19
𝐴𝑤 = =
2 7 −5 −31
𝐴𝑤 = (−19) 2 + (−31)2 = 1322

8. 𝑑 𝑢, 𝑤 = 𝑢 − 𝑤 = 𝑢 − 𝑤, 𝑢 − 𝑤 1/2 = −1 − 3 2 + 3−5 2 = 20
𝑑 𝑢, 𝑤 = 𝑢 − 𝑤 = 𝑢 − 𝑤, 𝑢 − 𝑤 1/2
= 5 −1 − 3 2 +2 3−5 2 = 88
𝑢 − 𝑤 = −1,3 − 3,5 = −4, −2
−2 3 −4 2
𝐴∙ 𝑢− 𝑤 = =
2 7 −2 −22
𝐴∙ 𝑢− 𝑤 = (2) 2 + (−22)2 = 488

9. 1/2
1/2 2 4 2 4
𝐴 = 𝐴, 𝐴 = , = 22 + 42 + (−3)2 +1 = 30
−3 1 −3 1
𝑑 𝐴, 𝐵 = 𝐴 − 𝐵 = 𝐴 − 𝐵, 𝐴 − 𝐵 1/2
= (2 + 4)2 +(4 − 2)2 +(−3 − 5)2 + 1 − 1 = 104
1/2 −1 6 −1 1/2
6
𝐴 = 𝐴, 𝐴 = , = 62 + (−1)2 +72 + 42 = 102
4 7 7 4
𝑑 𝐴, 𝐵 = 𝐴 − 𝐵 = 𝐴 − 𝐵, 𝐴 − 𝐵 1/2 = (6 + 1)2 +(−1 − 8)2 +72 + (4 − 2)2
= 183

10. 1/2
1 1
1/2
𝑝 = 𝑝, 𝑝 = 𝑝 𝑥 𝑝 𝑥 𝑑𝑥 = 3𝑥 2 − 2 3𝑥 2 − 2 𝑑𝑥
0 0
1 1
9𝑥 5 12𝑥 3 9
= 9𝑥 4 − 12𝑥 2 + 4 𝑑𝑥 = − + 4𝑥 =
0 5 3 0
5
𝑝 − 𝑞 = 2𝑥 2 − x − 2
𝑑 𝑝, 𝑞 = 𝑝− 𝑞
1/2
1
1/2
= 𝑝 − 𝑞, 𝑝 − 𝑞 = (2𝑥 2 − x − 2) 2𝑥 2 − x − 2 𝑑𝑥
0
1
52
= 4𝑥 4 − 4𝑥 3 − 7𝑥 2 + 4𝑥 + 4 𝑑𝑥 =
0 15

11. 1/2
1 1
1/2
𝑝 = 𝑝, 𝑝 = 𝑝 𝑥 𝑝 𝑥 𝑑𝑥 = 𝑥2 + 𝑥 + 1 𝑥 2 + 𝑥 + 1 𝑑𝑥
0 0
1 1
𝑥 5 2𝑥 4 3𝑥 3 2𝑥 2 37
= 𝑥4 + 2𝑥 3 + 3𝑥 2 + 2𝑥 + 1 𝑑𝑥 = + + + + 𝑥 =
0 5 4 3 2 0
10
𝑝 − 𝑞 = −4𝑥 2 + 2x − 2
𝑑 𝑝, 𝑞 = 𝑝− 𝑞
1/2
1
1/2
= 𝑝 − 𝑞, 𝑝 − 𝑞 = (−4𝑥 2 + 2x − 2 ) −4𝑥 2 + 2x − 2 𝑑𝑥
0
1
16𝑥 5 16𝑥 4 20𝑥 3 8𝑥 2 88
= − + − + 4𝑥 =
5 4 3 2 0
15

12. 2𝑢, 𝑣 − 𝑤 + 𝑣, 𝑣 − 𝑤 = 2𝑢, 𝑣 − 2𝑢, 𝑤 + 𝑣, 𝑣 − 𝑣 − 𝑤
= 2 𝑢, 𝑣 − 2 𝑢, 𝑤 + 𝑣 2 − 𝑣 − 𝑤 = 2 3 − 2 7 + 25 + 2 = 19
𝑢, 𝑣 + 3𝑤 − 𝑣, 𝑣 + 3𝑤 + 2𝑤, 𝑣 + 3𝑤
= 𝑢, 𝑣 + 𝑢, 3𝑤 − 𝑣, 𝑣 − 𝑣, 3𝑤 + 2𝑤, 𝑣 + 2𝑤, 3𝑤
= 3 + 3 7 − 52 − 3(−2) + 2(−2) + 6(8)2 = 385
1/2
2𝑢 + 𝑤, 2𝑢 + 𝑤 = 2𝑢, 2𝑢 + 𝑤 + 𝑤, 2𝑢 + 𝑤 1/2
= 4 𝑢, 𝑢 + 2 𝑢, 𝑤 + 2 𝑤, 𝑢 + 𝑤, 𝑤 1/2
= 4(2)2 + 2(7) + 2(7) + 82 1/2 = 108
𝑢 − 3𝑣 + 𝑤, 𝑢 − 3𝑣 + 𝑤 1/2
= 𝑢 , 𝑢 − 3𝑣 + 𝑤 − 3𝑣, 𝑢 − 3𝑣 + 𝑤 + 𝑤, 𝑢 − 3𝑣 + 𝑤 1/2
= ( 𝑢, 𝑢 − 3 𝑢, 𝑣 + 𝑢, 𝑤 − 3 𝑣, 𝑢 + 9 𝑣, 𝑣 − 3 𝑣, 𝑤 + 𝑤, 𝑢 − 3 𝑤, 𝑣
+ 𝑤, 𝑤 )1/2
= (2)2 −3 3 + 7 − 3 3 + 9 5 2 − 3 −2 + 7 − 3(−2) + 82 1/2 = 301

13. 𝑢, 𝑘𝑣 = 𝑘𝑣, 𝑢 symmetry 𝑢 − 𝑣, 𝑤 = 𝑤, 𝑢 − 𝑣 symmetry
= 𝑘 𝑣, 𝑢 homogeneity = 𝑤, 𝑢 + 𝑤, −𝑣 additivity
= 𝑘 𝑣, 𝑢 symmetry = 𝑤, 𝑢 + (−1) 𝑤, 𝑣 homogeneity
= 𝑢, 𝑤 − 𝑣, 𝑤 symmetry

14. 𝑢, 𝑣 = 5𝑢1 𝑣1 + 2𝑢2 𝑣2 = 5 2 0 + 2 3 −1 = −6 = 6 = 36
𝑢 = 𝑢, 𝑢 1/2 = 5(2)2 + 2(3)2 = 38
1/2
𝑣 = 𝑣, 𝑣 = 5(0)2 + 2(−1)2 = 2
compare
𝑢 ∙ 𝑣 = 38 2 = 76
36 < 76
∴ 𝑢, 𝑣 ≤ 𝑢 ∙ 𝑣
Cauchy-Schwarz inequality

15. 2 6 −3 1
𝑢, 𝑣 = , = −6 + 6 + 4 − 6 = −2 = 2 = 4
1 −3 4 2
2 6 2 6
𝑢 = 𝑢, 𝑢 1/2 = , = 4 + 36 + 1 + 9 = 50
1 −3 1 −3
−3 1 −3 1
𝑣 = 𝑣, 𝑣 1/2 = , = 9 + 1 + 16 + 4 = 30 compare
4 2 4 2
𝑢 ∙ 𝑣 = 50 30 = 1500
4 < 1500
∴ 𝑢, 𝑣 ≤ 𝑢 ∙ 𝑣
Cauchy-Schwarz inequality

16. • Do as your exercise

17. 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 1.5.3(𝑏)
𝑣 =0 iff 𝑣 = 0
𝑣 = 𝑣, 𝑣 1/2 = 𝑣1 2 + 𝑣2 2 + ⋯ + 𝑣 𝑛 2 ≥ 0
𝑣 = 𝑣, 𝑣 1/2 = 𝑣1 2 + 𝑣2 2 + ⋯ + 𝑣 𝑛 2 = 0 iff 𝑣1 = 𝑣2 = ⋯ = 𝑣 𝑛 = 0 → 𝑣 = 0
∴ 𝑣 =0 iff 𝑣 = 0
𝑇ℎ𝑒𝑜𝑟𝑒𝑚 1.5.3(𝑐)
𝑘𝑣 = 𝑘 𝑣
𝑘𝑣 = 𝑘𝑣, 𝑘𝑣 1/2 = 𝑘 𝑣, 𝑘𝑣 1/2 = 𝑘 2 𝑣, 𝑣 1/2 = 𝑘 𝑣, 𝑣 1/2 = 𝑘 𝑣

18. 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 1.5.3(𝑓)
𝑑 𝑢 , 𝑣 =0 iff 𝑢 = 𝑣
𝑑 𝑢, 𝑣 = 𝑢 − 𝑣 = 𝑢 − 𝑣, 𝑢 − 𝑣 1/2
= 𝑢1 − 𝑣1 2 + 𝑢2 − 𝑣2 2 + ⋯+ 𝑢 𝑛 − 𝑣𝑛 2 ≥0
𝑑 𝑢, 𝑣 = 𝑢− 𝑣 = 𝑢1 − 𝑣1 2 + 𝑢2 − 𝑣2 2 +⋯+ 𝑢 𝑛 − 𝑣𝑛 2 = 0 iff
𝑢1 = 𝑣1 , … 𝑢 𝑛 = 𝑣 𝑛 → 𝑢 = 𝑣
∴ 𝑑 𝑢, 𝑣 =0 iff 𝑢 = 𝑣
𝑇ℎ𝑒𝑜𝑟𝑒𝑚 1.5.3(ℎ)
𝑑 𝑢, 𝑣 ≤ 𝑑 𝑢, 𝑤 + 𝑑 𝑤, 𝑣
𝑑 𝑢, 𝑣 = 𝑢− 𝑣 = 𝑢+ 𝑤− 𝑤− 𝑣 = 𝑢− 𝑤 + 𝑤− 𝑣
≤ 𝑢− 𝑤 + 𝑤− 𝑣
= 𝑑 𝑢, 𝑤 + 𝑑 𝑤, 𝑣
∴ 𝑑 𝑢, 𝑣 ≤ 𝑑 𝑢, 𝑤 + 𝑑 𝑤, 𝑣

19. -dr Radz