Hi there I have a linear algebra homework question and I don

Hi there,

I have a linear algebra homework question and I don\'t understand what the question is asking:

Describe geometrically the following sets as points, lines, planes, or all of Rⁿ:

1) (a) x1 x2 : 2x1 + 3x2 = 1, x1 + x2 = 3

(b) span (1, 1, 2 ), (0, 1, 1 ) , (1 0 2 )

(c) span (1, 1, 2) , ( 0, 1, 4 ) , (1, 0, 2 )

Solution

Question wants you to describe how the graph of the set of given points would look like

a)

2x1+3x2=1

-x1+x2=3

Adding 2 times second equation to first gives

5x2=7

x2=7/5

-x1+x2=3

x1=x2-3=7/5-3=-8/5

Hence set of points is: (-8/5,7/5) which is a point in R2

b)

Let r,s,t so that

r(1,1,2)+s(0,1,-1)+t(-1,0,2)=0

r-t=0 giving,t=r

r+s=0 giving, s=-r

2r-s+2t=0

2r+r-2r=r=0 so

So, r=0=s=t

Hence set of vectors in linearly independent in R3

Hence they span whole of R3

Hence geometrically span is R3

c)

-(1,1,2)+(0,1,4)=(-1,0,2)

ie first two vectors span the third vector

And first two vectors together give a plane ie their span gives a plane.