MAT 242 Chap.14 Linear Transformations and Their Matrices
Linear Transformations and Their Matrices
Functions that respect linear combinations
A linear transformation sends sums to sums and scalar multiples to scalar multiples. Every linear transformation between spaces of column vectors is multiplication by a matrix, and the columns of that matrix are the images of the standard basis vectors.
The syllabus lists finding that matrix, and bases for the kernel and image, among the course learning outcomes.
Images of a non-standard basis
Often the transformation is given by where it sends some other basis. Put those input vectors into the columns of M and their images into the columns of N; then AM = N, so A = NM⁻¹.
The inverse is essential here, and the sample solutions deduct for using M itself in place of M⁻¹. A quick check multiplies A by each input vector and recovers the stated image.
Kernel and image
The kernel, the inputs sent to zero, is the null space of the matrix; the image, or range, is its column space.
So the Chapter 8 procedures apply unchanged, with one extra requirement: label which basis is which, since they live in different spaces.
The paid chapter works both computations and illustrates a geometric transformation, a shear of the unit square.
The course sources differ on whether this material is on the Final Exam, so confirm the scope on Canvas; the calendar still lists online assignments for these sections.
How the sample final grades this
The sample final's transformation questions give credit for writing M, writing N and computing A, and they deduct for multiplying by M instead of by its inverse.
The kernel-and-image question gives credit for the reduction and for each basis, and deducts when the two bases are not labelled.
The course calendar also lists an online assignment on geometric transformations, such as the shear pictured in the paid chapter, where the matrix again comes from the images of the standard basis vectors.
Geometric transformations
Many familiar plane motions are linear: rotations about the origin, reflections in lines through the origin, stretches along an axis and shears.
Each is determined by where it sends the two standard basis vectors, and those images are the columns of its matrix. Translations are not linear, because they move the origin, which is a quick test for spotting a non-linear map. Composing two linear maps multiplies their matrices, with the map applied first on the right.
What this chapter covers
- 01
The definition of a linear transformation
- 02
Columns as images of basis vectors
- 03
A = NM⁻¹ from images of any basis
- 04
Kernel as a null space
- 05
Image as a column space
Worked example · free
Matrix, kernel and image of a map from R² to R³
- 2Images of (1, 0) and (0, 1) are (1, 2, 0) and (1, 0, 1), so A has those columns.
- 2The RREF of A has a pivot in both columns, so the kernel is only the zero vector.
- 1The image has basis {(1, 2, 0), (1, 0, 1)}, the two columns of A.
Key terms
- Linear Transformation
- A linear transformation is a function that preserves vector addition and scalar multiplication.
- Kernel
- The kernel of L is the set of inputs that L sends to the zero vector, the null space of its matrix.
- Image
- The image of L is the set of all its outputs, the column space of its matrix.
Linear Transformations and Their Matrices FAQ
How do I recognise that a function is not linear?
Check where it sends the zero vector and whether it respects sums. A shift such as adding a fixed nonzero vector sends zero somewhere else, so it cannot be linear.
Why is the matrix N times M inverse rather than M inverse times N?
The defining relation is A times M equals N, because A acts on each input column of M. Solving for A means multiplying on the right by M inverse.
How do I find the matrix of a transformation from its formula?
Apply the formula to each standard basis vector in turn. The resulting output vectors, written as columns in order, form the matrix of the transformation.
Exam move
Treat this chapter as Chapter 8 in new clothing. Practise one matrix-from-images problem and one kernel-and-image problem, and confirm the Final Exam scope before investing more time. Reuse Chapter 8 practice matrices here: read each one as a transformation and name its kernel and image to connect the two chapters.
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