TITLE: Linear independency before and after Linear Transformation
QUESTION [20 upvotes]: If we are given some linearly dependent vectors, would the T of those vectors necessarily be dependent (given a transformation from $R^n$ to $R^p$)?
And if we are given some linearly independent vectors, would T of those vectors necessarily be independent (given a transformation from $R^n$ to $R^p$)?
The answers are presumably no and no, but I am struggling to figure out why. Thanks!

REPLY [2 votes]: Just to add to Arturo Magidin's answer, if you are considering sets of vectors rather than lists, then both statements are false:
Consider $T:\mathbb{R}^2 \to \mathbb{R}, \: T(x,y) = x+y,$ and sets $A = \{(2,0),(0,2),(1,1)\}$ and $B = \{(1,-1)\}$. Then $A$ is linearly dependent, but $T(A) = \{2\}$ is linearly independent; and $B$ is linearly independent, but $T(B) = \{0\}$ is linearly dependent.