Final answer:
A linear transformation is a function from one vector space to another that preserves both scalar multiplication and vector addition.
Step-by-step explanation:
A linear transformation is a function from one vector space to another that preserves both scalar multiplication and vector addition. In other words, if T is a linear transformation from vector space V to vector space W, then for any scalar c and any vectors u and v in V, the following properties must hold:
T(cu) = cT(u) (preservation of scalar multiplication)
T(u + v) = T(u) + T(v) (preservation of vector addition)
For example, let's consider the transformation T: R^2 -> R^2 defined by T(x, y) = (3x - 2y, 2x + y). We can verify that T satisfies both properties:
T(c(x, y)) = T(cx, cy)
= (3(cx) - 2(cy), 2(cx) + (cy))
= (3cx - 2cy, 2cx + cy)
= c(3x - 2y, 2x + y)
= cT(x, y)
T((x, y) + (a, b))
= T(x + a, y + b)
= (3(x + a) - 2(y + b), 2(x + a) + (y + b))
= (3x + 3a - 2y - 2b, 2x + 2a + y + b)
= (3x - 2y, 2x + y) + (3a - 2b, 2a + b)
= T(x, y) + T(a, b)