Linear Algebra Chapter 1.8 Quiz

Linear Transformations

• A linear transformation maps vectors from R^n to ℝ^m, assigning each vector x in R^n to vector T(x) in ℝ^m.
• For a transformation defined by T(x) = Ax, if A is a 3×5 matrix, the domain of T is R^5, not R^3. The input vector x must match the second dimension of matrix A.

Range and Domain

• The range of the transformation x ↦ Ax is not necessarily ℝ^m; instead, it consists of all linear combinations of the columns of matrix A.
• A matrix transformation is a specific type of linear transformation represented as x ↦ Ax, establishing that not all linear transformations are matrix transformations.

Properties of Linear Transformations

• A transformation T is linear if it meets the condition T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all vectors v1, v2, and scalars c1, c2 in the domain.
• Linear transformations preserve vector addition and scalar multiplication, demonstrated by T(cu + dv) = cT(u) + dT(v).

Specific Characteristics

• The range of the transformation x ↦ Ax includes all linear combinations of the columns of A, confirming a link between the transformation and the structure of matrix A.
• Every matrix transformation is indeed a linear transformation, following the properties of linearity in T.

Existence and Uniqueness Questions

• In context of linear transformations T: R^n → R^m, questioning if a vector c is in the range of T pertains to existence (whether there is an x such that T(x) = c) rather than uniqueness.

Origin Mapping

• A linear transformation T: R^n → R^m guarantees that the origin of R^n (T(0)) is always mapped to the origin of R^m (0), affirming the structure-preserving nature of linear transformations.