Exercise 11. Let L : R^4 - > R^3 be the map defined as L(x,y,t, u) = (t - 2x - 3y + u, 4x + 6y -
2u, t) (1) Prove that L is linear: (2) Compute dim (Im(L)). Hint: use the fact that dim(Im(L)) is
the column rank of AL) (3) Compute dim(ker(L)). (Hint: use part (2) together with the
Dimension Formula.)
Solution
For proving T is linear let us show that L(x1+x2, y1+y2, t1+t2, u1+u2) = L(x1,y1,
z1,t1)+L(x2,y2, z2,t2)
LHS = (t1+t2-2x1-2x2-3y1-3y1+u1+u2, ......)
By property of distributivity and commutativity for real numbers we get LHS =RHS
Hence t is linear.
2) dim (IML) = 3 as there are 4 linearly independent vectors
3) Ker(L) = (set of (x,y,t,u) such that
t-2x-3y+u =0
4x+6y-2u =0 and t =0
i.e. 2x+3y+u =0 and t =0 is the condition
This has solution as
(x,y, 0,-2x-3y)
Dim ker L = 2..
Exercise 11. Let L R^4 - R^3 be the map defined as L(x,y,t, u) = .pdf
1. Exercise 11. Let L : R^4 - > R^3 be the map defined as L(x,y,t, u) = (t - 2x - 3y + u, 4x + 6y -
2u, t) (1) Prove that L is linear: (2) Compute dim (Im(L)). Hint: use the fact that dim(Im(L)) is
the column rank of AL) (3) Compute dim(ker(L)). (Hint: use part (2) together with the
Dimension Formula.)
Solution
For proving T is linear let us show that L(x1+x2, y1+y2, t1+t2, u1+u2) = L(x1,y1,
z1,t1)+L(x2,y2, z2,t2)
LHS = (t1+t2-2x1-2x2-3y1-3y1+u1+u2, ......)
By property of distributivity and commutativity for real numbers we get LHS =RHS
Hence t is linear.
2) dim (IML) = 3 as there are 4 linearly independent vectors
3) Ker(L) = (set of (x,y,t,u) such that
t-2x-3y+u =0
4x+6y-2u =0 and t =0
i.e. 2x+3y+u =0 and t =0 is the condition
This has solution as
(x,y, 0,-2x-3y)
Dim ker L = 2.
