- #1
DanielFaraday
- 87
- 0
Homework Statement
If [tex]T:U\rightarrow V[/tex] is any linear transformation from U to V and [tex]B=\left\{u_1,u_2,\text{...},u_n\right\}[/tex] is a basis for U, then the set [tex]T(B)=\left\{T\left(u_1\right),T\left(u_2\right),\text{...} T\left(u_n\right)\right\}[/tex]
a. spans V.
b. spans U.
c. is a basis for V.
d. is linearly independent.
e. spans the range of T.
Homework Equations
None
The Attempt at a Solution
It seems to me like all of these things are true (which is wrong of course). But in the example below, the result of the transformation does in fact meet all of the above criteria. What am I missing here?
[tex]
u_1=\left(
\begin{array}{c}
1 \\
0 \\
0
\end{array}
\right)
[/tex]
[tex]
u_2=\left(
\begin{array}{c}
0 \\
1 \\
0
\end{array}
\right)
[/tex]
[tex]
u_3=\left(
\begin{array}{c}
0 \\
0 \\
1
\end{array}
\right)
[/tex]
[tex]
T(x,y,z)=(x,2y,3z)
[/tex]
[tex]
T\left(u_1\right)=\left(
\begin{array}{c}
1 \\
0 \\
0
\end{array}
\right)
[/tex]
[tex]
T\left(u_2\right)=\left(
\begin{array}{c}
0 \\
2 \\
0
\end{array}
\right)
[/tex]
[tex]
T\left(u_3\right)=\left(
\begin{array}{c}
0 \\
0 \\
3
\end{array}
\right)
[/tex]