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