Let L(x1; x2; x3; x4; x5; x6; x7) = 2x3 + 3x4 + 8x6 + x7: Is L a linear map? If it is, find as many independent vectors in Ker(L) as you can. Solution L(x1; x2; x3; x4; x5; x6; x7) = 2x3 + 3x4 + 8x6 + x7 Since L satisfies scalar multipication and addition properties, it is a LINEAR map rank of transformation matrix = 4 =>dimension of kernal = 7-4 = 3 =>maximum number of independent vectors in kernel = 3 =>these vectors are : (1 0 0 0 0 0 0), ( 0 1 0 0 0 0 0) , (0 0 0 0 1 0 0 ) .

1. Let L(x1; x2; x3; x4; x5; x6; x7) = 2x3 + 3x4 + 8x6 + x7: Is L a linear map?
If it is, find as many independent vectors in Ker(L) as you can.
Solution
L(x1; x2; x3; x4; x5; x6; x7) = 2x3 + 3x4 + 8x6 + x7
Since L satisfies scalar multipication and addition properties, it is a LINEAR map
rank of transformation matrix = 4
=>dimension of kernal = 7-4 = 3
=>maximum number of independent vectors in kernel = 3
=>these vectors are : (1 0 0 0 0 0 0), ( 0 1 0 0 0 0 0) , (0 0 0 0 1 0 0 )