Linear Transformations, Part 1

Linear Transformations

Part 1: Matrix Transformations

An m x n matrix A defines a function T from Rⁿ to R^m by taking a vector x in Rⁿ to the vector T(x) = Ax in R^m.

Let T: R⁵ --> R³ be defined by T(x) = Ax. Enter the matrix A and the vectors u and v defined in your worksheet. Compute

T(u), T(v) and T(u) + T(v)

u + v and T(u + v)

What do you hypothesize about the relationship between T(u) + T(v) and T(u + v)? Prove your statement (for any m x n matrix A and any vectors u, v in Rⁿ) using matrix algebra.

Compute 3T(u) and T(3u). Repeat these computations for v. Then repeat them with 3 replaced by -2/5. What do you deduce? Prove your statement (for any m x n matrix A, any vector u in Rⁿ, and any scalar c) using matrix algebra.

Suppose that S: R² --> R⁴ is defined via multiplication by the matrix B. What is the size of the matrix B?

Assume that S is the matrix transformation discussed in the previous step. If S((1, 0)) = (1, 0, -2, 4) and S((0, 1)) = (-3, 5, 1, 2), find

S((2, 0))
S((1, 1))
S((-3, 2))
the matrix B

A matrix map, such as T or S defined above, is a special case of a more general type of map called a linear transformation. A linear transformation from a vector space V to a vector space W is a function with domain V and target W that satisfies preserves the properties of addition and scalar multiplication. More precisely, a function T: V --> W is a linear transformation provided that, for all vectors u and v in V and for all scalars c, the following two properties hold:

T(u + v) = T(u) + T(v) (preservation of addition), and

T(cu) = cT(u) (preservation of scalar multiplication).

Let V be the vector space of real polynomials of degree less than or equal to 3, and let W be the vector space of real polynomials of degree less than or equal to 2. Let T: V --> W be defined by "taking the derivative"; i.e., if

p(x) = a + bx + cx² + dx³,

then

T(p)(x) = b + 2cx + 3dx².

Show that T is a linear transformation.