Let T: R²→ R³ be the linear transformation defined by T(x, y) = (3x − y, 5x + 5y, y — 5x). Find a vector w that is not in the image of T.

Answer 1

Hi! This question is well-defined. Why? Because any linear transformation from R² into R³ cannot be onto. This means that we can always find some vector (infinitely many indeed) w in R³ that is not the image of T.

There can be many ways to find vectors that do not belong to the image of T... one of them is the following:

For example, the vector (0,0,1) cannot be written as T(x,y)=(0,0,1) because the system 3x-y=0, 5x+5y=0, y-5x=1 has no solutions (in linear algebra one could say that this is true because the augmented matrix corresponding to the system has a pivot position in the right-most column.

I hope this helps! Good luck.