Answer to Question #254029 in Linear Algebra for usama

Question #254029

Show that the vectors u1 = (1, 1, 1), u2 = (1, 2, 3), u3 = (1, 5, 8) span R

3

.


1
Expert's answer
2021-10-22T01:16:22-0400

Solution:

To solve it show that an arbitrary vectorsl v=(a,b,c) \in R3 is a linear combination of u1,u2u_1,u_2 and u3u_3 .

Let

v=xu1+yu2+zu3v=xu_1+yu_2+zu_3

Hence,

(a,b,c)=x(1,1,1)+y(1,2,3)+z(1,5,8)(a,b,c)=x(1,1,1)+y(1,2,3)+z(1,5,8)

Resolve as

x+y+z=ax+y+z=a

x+2y+5z=bx+2y+5z=b

x+3y+8z=cx+3y+8z=c

Form the equivalent matrix form of the equations

[111a125b138c]\begin{bmatrix} 1&1& 1&\vert&a \\ 1&2 & 5&\vert&b\\ 1&3&8&\vert&c \end{bmatrix}

Reduced echelon form is

[111a011ba001a2b+c]\begin{bmatrix} 1 & 1&1&\vert&a\\ 0 & 1&1&\vert&b-a\\ 0&0&-1&\vert&a-2b+c \end{bmatrix}

From the reduced echelon form,

z=a2b+c-z=a-2b+c

z=a+2bcz=-a+2b-c

y+4z=bay+4z=b-a

y=ba4zy=b-a-4z

By substitution,

y=ba4(a+2bc)y=b-a-4(-a+2b-c)

y=3a7b+4cy=3a-7b+4c

Also from the echelon form,

x+y+z=ax+y+z=a

x=ayzx=a-y-z

By substitution,

x=a(3a7b+4c)(a+2bc)x=a-(3a-7b+4c)-(-a+2b-c)

x=a+5b3cx=-a+5b-3c

Therefore,

(a,b,c)=(a+5b3c)(1,1,1)+(3a7b+4c)(1,2,3)+(a+2bc)(1,5,8)(a,b,c)=(-a+5b-3c)(1,1,1)+(3a-7b+4c)(1,2,3)+(-a+2b-c)(1,5,8)

Therefore,

u1,u2,u3u_1,u_2,u_3 span R3R^3













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