Let V and W be vector spaces and T V rightarrow W be a linea

Let V and W be vector spaces, and T: V rightarrow W be a linear transformation. Let H = {v_1, v_2, ..., v_p} be a set of vectors from V. Show that if H is linearly dependent, then T(H) = {T(v_1), T(v_2), ...., T(v_p)} is also linearly dependent.

Solution

as given let V and W be vector spaces and T : V -> W be a linear transformation.

Let H = {v₁ , v₂ , . . . , v_p} be a set of vectors from V.

we need to show that if H is linearlt dependent then T(H) = {T(v1) , T(v2) , . . . , T(v_p)} is also linearly dependent.

we know that If H is linearly dependent then there exists the coefficients c₁ , c₂ , . . . , c_p such that,

c₁v₁ + c₂v₂ + . . . + c_pv_p = 0

where c₁ , c₂ , . . ., c_p not all of them are zero

now applying linear trasformation T on both side we have,

T(c₁v₁ + c₂v₂ + . . . + c_pv_p) = 0_W

T(c₁v₁) + T(c₂v₂) + . . . + T(c_pv_p) = 0_W

c₁T(v₁) + c₂T(v₂) + . . . + c_pT(v_p) = 0_W

which means that there exist c₁ , c₂ , . . . , c_p such that linear combination of transformed vector is zero vector 0_W

hence we can say that T(H) = {T(v1) , T(v2) , . . . , T(v_p)} is linearly dependent.

where c₁ , c₂ , . . ., c_p not all of them are zero