Final answer:
To find a basis for the set of vectors in R3 in the given plane x1+5x2−2x3=0, we can express the equation in vector form and identify the vectors that can form a basis. In R4, we can do the same for the three given subspaces and find the vectors that form the bases.
Step-by-step explanation:
To find a basis for the set of vectors in R3 in the plane x1+5x2−2x3=0, we can solve the equation and express it in vector form. The equation x1+5x2−2x3=0 can be rewritten as x1(1,0,0)+x2(0,1,0)-x3(0,0,2)=0. Therefore, a basis for this plane is {(1,0,0), (0,1,0), (0,0,2)}.
To find a basis for the set of vectors in R4 in the subspace (hyperplane) A) X1-2X2-X3+6X4=0, we can solve the equation and express it in vector form. Expressing the equation in vector form gives X1(1,0,0,0)-X2(0,2,0,0)-X3(0,0,1,0)+X4(0,0,0,6)=0. Therefore, a basis for this hyperplane is {(1,0,0,0), (0,2,0,0), (0,0,1,0), (0,0,0,6)}.
A basis for the set of vectors in R4 in the subspace (hyperplane) B) 3X1-10X2+X3+6X4=0 is {(3,-10,1,6)}.
A basis for the set of vectors in R4 in the subspace (hyperplane) C) X1-4X2+X3=0 is {(1,-4,1,0)}.