How do you find a basis for the given subspaces of $\displaystyle R^3$ and $\displaystyle R^4$.
a. All vectors of the form (a,b,c), where a=0
b. All vectors of the form (a+b, a-b, b+c, -a+b)
c. All vectors of the form (a,b,c), where a-b+5c=0
How do you find a basis for the given subspaces of $\displaystyle R^3$ and $\displaystyle R^4$.
a. All vectors of the form (a,b,c), where a=0
b. All vectors of the form (a+b, a-b, b+c, -a+b)
c. All vectors of the form (a,b,c), where a-b+5c=0
$\displaystyle (0,b,c) = (0,b,0) + (0,0,c) = b(0,1,0) + c(0,0,1)$
$\displaystyle (a,a,0,-a) + (b,-b,b,b) + (0,0,c,0) = a(1,1,0,-1) + b(1,-1,1,1) + c(0,0,1,0)$b. All vectors of the form (a+b, a-b, b+c, -a+b)
$\displaystyle (a,b,c) = (a,a+5c,c) = a(1,1,0) + c(0,5,1)$.c. All vectors of the form (a,b,c), where a-b+5c=0