Let F be a field, U and V vector spaces above field F and let T:V-->W be a linear transformation.
Prove or disprove the next statements:
1. If v_1, v_2, ... , v_n are vectors in vector space V and they are independent linear, then so T(v_1), ...T(v_n) independent linear.
2. If v_1, v_2, ... , v_n are vectors in vector space V and they are dependent linear, then so T(v_1), ...T(v_n) dependent linear.
I hope you understood my question, eve thou my broken English.
I will appreciate a full answer!
December 10th 2009, 07:19 AM
the first statement is false. A couterexample is T=0 for all vectors in V.
the second statement is true. Because a linear transformation preserve addition and multiplication by scalar, so if v_1,...v_n are linear dependent,that is, there exist nubmers a_1,...a_n (not all 0) in F such that the linear combination of v_1,...v_n is 0, thus the image of this linear combination under T is 0 in W, that is T(v_1), ...T(v_n) is dependent linear.
To summarize, A linear transform T preserve the linear independency iff T is injective( T is invertable).