Find a basis for each of these subspaces of R4: All vectors whose components arc

Question

Find a basis for each of these subspaces of R4: All vectors whose components arc equal. All vectors whose components add to zero. All vectors that are perpendicular to (1, 1, 0, 0) and (1,0, 1, 1). The column space (in R2) and nullspace (in R5) of U =

Explanation / Answer

(A) consider a vector (u,v,w,x) in R4 given u=v=w=x => this vector can be written as k(1,1,1,1) where k is a scalar Hence basis = {(1,1,1,1)} (B) consider a vector (u,v,w,x) in R4 given u+v+w+x = 0 => u = -v-w-x => this vector can be written as v(-1,1,0,0)+w(-1,0,1,0)+x(-1,0,0,1) Hence basis = {(-1,1,0,0),(-1,0,1,0),(-1,0,0,1)} (C) consider a vector (u,v,w,x) in R4 given u+v = 0 u+w+x = 0 => v = -u x = -u-w => this vector can be written as u(1,-1,0,-1)+w(0,0,1,-1) Hence basis = {(1,-1,0,-1),(0,0,1,-1)} (D) basis of column space = {(1,0),(0,1)} Let null space contains vector (u,v,w,x,y) of R5 as it is in null space, u+w+y = 0 v+x = 0 => u = -w-y v = -x => this vector can be written as w(-1,0,1,0,0)+x(0,-1,0,1,0)+y(-1,0,0,0,1) Hence basis of nullspace = {(-1,0,1,0,0),(0,-1,0,1,0),(-1,0,0,0,1)}

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