Linear Transformations

Question 1

General Properties of Linear Transformations

Answer 1

It can be shown that all linear transformations satisfy certain basic properties.
* For a linear transformation T from a vector space V to a vector space W ,
two such properties are:

(1) T (0) = 0 (where the 0’s on the LHS & RHS are those of V & W , respectively).

(2) T (−u) = −T (u) for all u ∈ V (where the negatives on the LHS & RHS are
those in V & W , respectively)

Question 2

Linear Transformation

Answer 2

Linear Transformations
* Let V & W be vector spaces.
* A linear transformation from V to W is a defined to be a function T : V → W
which satisfies the following two conditions.

(a) T (u + v) = T (u) + T (v) for all u,v ∈ V .

(b) T (k u) = k T (u) for all k ∈ R & all u ∈ V