# Kernel/Range basis

by newtomath
Tags: basis, kernel or range
 P: 37 L: R^4 => R^3 is defined by L(x,y,z,w) = (x+y, z+w, x+z) A) Find a basis for ker L We can re write L(x,y,z,w) as x* (1,01) + y *(1,0,0) + z*(0,1,1) + w*(0,1,0). I then reduced it to row echelon form We now have the equations X-W=0 , Y+W=0, Z+W=0. There are infinitely many solutions as X=W, Y= -W and Z=-W. So if we set W=1 we have the basis for the kernel=Vector(1,-1, -1,1) B) find a basis for range L Given L(x,y,z,w) = (x+y, z+w, x+z) We can re write L(x,y,z,w) as x* (1,01) + y *(1,0,0) + z*(0,1,1) + w*(0,1,0). S= {(1,01) ,(1,0,0) ,(0,1,1),(0,1,0)} It spans L. To find the basis for L we set {x* (1,01) + y *(1,0,0) + z*(0,1,1) + w*(0,1,0)} = 0,0,0 I reduced it and the leading one's appear in the first 3 columns of the reduced form, the first 3 vectors in the original matrix became a basis for the range of L They are: Vector( {(1, (0, (1}) , Vector( 1, 0, 0}) , Vector( 0, 0, 1}) C) verify theorem 10.7 (dim(KerL) + dim(rangeL) = dim V The dim can be viewed as the # of vectors in of the Ker/range. Given (dim(KerL) + dim(rangeL) = dim V we have 1+3=4, which is the number of dimensions in the original space (L(x,y,z,w)).
Mentor
P: 20,440
 Quote by newtomath L: R^4 => R^3 is defined by L(x,y,z,w) = (x+y, z+w, x+z) A) Find a basis for ker L We can re write L(x,y,z,w) as x* (1,01) + y *(1,0,0) + z*(0,1,1) + w*(0,1,0). I then reduced it to row echelon form We now have the equations X-W=0 , Y+W=0, Z+W=0. There are infinitely many solutions as X=W, Y= -W and Z=-W. So if we set W=1 we have the basis for the kernel=Vector(1,-1, -1,1) B) find a basis for range L Given L(x,y,z,w) = (x+y, z+w, x+z) We can re write L(x,y,z,w) as x* (1,01) + y *(1,0,0) + z*(0,1,1) + w*(0,1,0). S= {(1,01) ,(1,0,0) ,(0,1,1),(0,1,0)} It spans L. To find the basis for L we set {x* (1,01) + y *(1,0,0) + z*(0,1,1) + w*(0,1,0)} = 0,0,0 I reduced it and the leading one's appear in the first 3 columns of the reduced form, the first 3 vectors in the original matrix became a basis for the range of L They are: Vector( {(1, (0, (1}) , Vector( 1, 0, 0}) , Vector( 0, 0, 1}) C) verify theorem 10.7 (dim(KerL) + dim(rangeL) = dim V The dim can be viewed as the # of vectors in of the Ker/range. Given (dim(KerL) + dim(rangeL) = dim V we have 1+3=4, which is the number of dimensions in the original space (L(x,y,z,w)).