Help with linear algebra. Figure 3.1: Figures for Problem 3.8.6 Problem 3.8 6: E

Question

Help with linear algebra.

Figure 3.1: Figures for Problem 3.8.6 Problem 3.8 6: Express the line segment in Figure 3.1(a) using a set of linear combinations. Do the same for the plane containing the triangle in Figure 3.1(b) Vector spaces Problem 3.8.7: Prove or give a counterexample: "t ar, y, 2 a, y, 2 e R, y 2 1 IS a vector space." Problem 3.8.8: Prove or give a counterexample ''tar, y, 2 ar, y, 2 E R and y 2 is a vector space." Problem 3.8.9: Prove or give a counterexample "1 a 1, T2, T3, T4, 5 1, T2, T3, T4, T5 E R, a2 0 or ar5 0 is a vector space

Explanation / Answer

Problem 3.8.6 Figure is not visible.

Problem 3.8.7

Consider two points in this set: (0,0,1) and (1,0,0)

Hence, (0,0,1)+(1,0,0)=(1,0,1) which is not in this set because: 1+0+1=2

Hence this set is not a vector space.

Problem 3.8.8

Denote this set by: S

1. (0,0,0) belongs to this set S

2. Let: (a,b,c) and (e,f,g) belong to the set S

(a,b,c)+(e,f,g)=(a+e,b+f,c+g)

a+e+b+f+c+g=(a+b+c)+(e+f+g)=0+0=0

Hence sum of any two elements in S belongs to S

3. Let c be a real number and (x,y,z) be in S

x+y+z=0

Hence, c(x+y+z)=0

cx+cy+cz=0

Hence, (cx,cy,cz) is also in S

Hence S is a vector space.

Problem 3.8.9

Consider two points in this space:

(1,0,1,1,1) and (1,1,1,1,0)

Note that in given space either second or fifth coordinate need to be zero. Both need not be zero.

Adding the two vectors gives:

(2,1,2,2,1) which is not in this space.

Hence this is not a vector space.

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