Linear Algebra · Chapter 08

Linear transformations

Linear maps between any two vector spaces: pictures of them in the plane, their kernels and images, the rank–nullity theorem, their matrices in any basis, change of basis, and every way of saying that a square matrix is invertible.

T (α u + β v) = α T (u) + β T (v)

ker (T)

im (T)

dim ker (T) + dim im (T) = dim V

[T]_{B}

T : V ⟶ W

B = P^{- 1} A P

x \mapsto A x

Lessons

Build the foundation

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06Change of basisThe same map looks different in different bases, and a well-chosen basis can make it look simple. Changing basis is a matter of translating coordinates in, doing the map, and translating back, and the three steps multiply into one matrix.Read lesson →

07The invertible matrix theoremThe course has met invertibility again and again in different clothes: a non-zero determinant, a pivot in every column, independent columns, a map with nothing in its kernel. For a square matrix these are all the same condition, and one of them holding makes every one of them hold.Read lesson →

08Chapter practiceBring the chapter together: recognise a linear map, find its kernel and image, count dimensions, write its matrix in a basis, change the basis, and decide invertibility. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Linear transformations

Build the foundation