# linearly dependent

• Dec 20th 2008, 12:24 PM
ssl000
linearly dependent
let T:R^n -> R^m be a linear transformation, and let {v1, v2, v3} be a linearly dependent set in R^n. Explain why the set {T(v1),(v2),(v2)} is linearly dependent.
• Dec 20th 2008, 05:37 PM
kalagota
first, the latter set contains two elements which are equal..

assuming my corrections below...
Quote:

Originally Posted by ssl000
let T:R^n -> R^m be a linear transformation, and let {v1, v2, v3} be a linearly dependent set in R^n. Explain why the set {T(v1),T(v2),T(v3)} is linearly dependent.

if \$\displaystyle \{v_1, v_2, v_3\}\$ is linearly dependent, then WLOG, \$\displaystyle v_1 = r_2v_2+ r_3v_3\$ where \$\displaystyle r_2\$ and \$\displaystyle r_3\$ are scalars..

\$\displaystyle T(v_1) = T(r_2v_2+ r_3v_3) = r_2T(v_2) + r_3T(v_3)\$ since \$\displaystyle T\$ is a linear transformation.
Thus you were able to express, WLOG, \$\displaystyle T(v_1)\$ as a linear combination of \$\displaystyle T(v_2)\$ and \$\displaystyle T(v_3)\$. Hence \$\displaystyle \{T(v1),T(v_2),T(v_3)\}\$