Linear Transformations

NOTE: Consider the vector spaces R and R2 over R.

T:V → V, where V is a vector space over R

T(x) = 0 ; x ∈ V

linear transformation

Guess:

T:R → R2

T(x) = (x, 3x); x ∈ R

linear transformation

Guess:

T:R2 → R2

T(x, y) = (x, 0); x, y ∈ R

linear transformation

Guess:

T:R2 → R2

T(x) = (1, 1); x ∈ R

linear transformation

Guess:

T:R → R2

T(x) = (x, 3x+1); x ∈ R

linear transformation

Guess:

T:R → R2

T(x) = (x, x2); x ∈ R

linear transformation

Guess:

Observation:

If T:R → R2 or T:R2 → R2 is linear, then Range T (i.e. Image T) is
or or