Expand the given set to form a basis for R2 0 4 Expand the g

Expand the given set to form a basis for R².

{[0 4]}

Expand the given set to form a basis for R².

{[2 1 3] , [3 2 6]}

Solution

1.An arbitrary element of R² is of the form [ a, b],where a,b R. Now,[ a, b]=a[1, 0]+b [ 0,1]=a/4[ 4, 0] + b/4[0, 4] so that the required basis for R² is { [ 0, 4] , [ 4,0]}

Note: The other element of the basis set can be [ p, 0 ] where p is a real number.

2. The set { [ 2,1,3], [ 3,2,6]} can be expanded to form a basis for R³ ( and not R²). ). Let A be the matrix with [ 2,1,3], [ 3,2,6] as columns,. Then the RREF of A is a matrix with columns [ 1,0,0] and [ 0,1,0] . Thus, the given vectors, together with the vector [ 0,0,1] will form a basis for R³ .