MAST10007 Chap.9 Image, Kernel, Rank–Nullity and Change of Basis
Image, Kernel, Rank–Nullity and Change of Basis
Chapter 9 develops image, kernel, rank–nullity and change of basis for University of Melbourne MAST10007. The kernel records inputs erased by a linear map; the image records outputs the map can actually reach. Rank–nullity balances their dimensions against the domain. Change of basis does not change the underlying vector, only its coordinate description.
A change matrix must be labelled by direction, and a reconstruction in standard coordinates is the fastest check against reversal. It pairs exact definitions with a new completed example, two concept diagrams, diagnostic contrasts and answered retrieval prompts.
The kernel and image answer complementary questions but live in different ambient spaces.
Kernel vectors are inputs that map to zero; image vectors are reachable outputs. Reduction reveals both through different readings, and rank–nullity checks that their dimensions account for the domain. Coordinate conversion adds another representation layer: the vector stays fixed while its coefficient list changes.
A reliable solution labels every basis and arrow, converts through a declared direction, and reconstructs the same standard-coordinate vector at both ends of the diagram.
What this chapter covers
- 01
Kernel
- 02
Image
- 03
One-to-one and the kernel
- 04
Onto and the image
- 05
Rank–nullity
- 06
Finding bases for image and kernel
- 07
Ordered bases
- 08
Change-of-coordinate matrices
- 09
Direction checks
- 10
Commuting coordinate diagrams
- 11
Domain and codomain labels for kernel and image
- 12
Rank–nullity as a dimension audit
- 13
Reconstructing both sides of a coordinate conversion
Image, Kernel, Rank–Nullity and Change of Basis worked example
- stepReconstruct v in standard coordinates: v=2(1,0)+3(1,1)=(5,3).
- stepWrite v=a(1,1)+b(0,1)=(a,a+b).
- stepMatching components gives a=5 and a+b=3, hence b=-2.
- stepTherefore [v]_C=(5,-2). Reconstructing 5(1,1)-2(0,1)=(5,3) checks that the change direction is B to C.
Key terms
- Kernel
- The kernel is every input sent to zero. It is found by solving Ax=0 and lives in the domain. A nonzero kernel vector records a lost direction and proves the transformation is not one-to-one.
- Image
- The image is every attainable output and equals the span of the matrix columns. It lies in the codomain. Pivot columns from the original matrix provide a basis, while rank gives its dimension.
- One-to-one and the kernel
- A linear map is one-to-one exactly when its kernel contains only zero. In matrix language, every variable column must pivot. This test focuses on inputs, not on whether the codomain has unused directions.
- Onto and the image
- A map is onto exactly when its image equals the codomain. For an m×n matrix, that requires a pivot in every row. Onto is therefore a statement about reaching all output directions.
- Rank–nullity
- Rank–nullity says dim(ker T)+dim(im T)=dim(domain). It balances lost input directions against surviving output directions. The right-hand side is the domain dimension, not the codomain dimension.
- Finding bases for image and kernel
- Kernel bases come from special solutions of Ax=0; image bases come from original pivot columns. Although both are revealed by the same reduction, they occupy different spaces and use different vectors.
- Rank–nullity audit
- Rank is the dimension of the image and nullity is the dimension of the kernel. Their sum equals the dimension of the domain. This relation checks whether the basis sizes obtained from pivot and free-variable calculations are mutually consistent.
- Change-of-coordinate matrix
- A change-of-coordinate matrix converts one ordered coefficient list into another for the same vector. Its columns are determined by the chosen source and destination bases, so a direction label is essential. Reversing direction requires the inverse conversion.
Image, Kernel, Rank–Nullity and Change of Basis FAQ
How do I check kernel?
The kernel is every input sent to zero. It is found by solving Ax=0 and lives in the domain. A nonzero kernel vector records a lost direction and proves the transformation is not one-to-one. Exactly when ker(T)={0}.
What is the main trap in image, kernel, rank–nullity and change of basis?
Kernel lives in the domain; image lives in the codomain. A change matrix with columns from basis B usually maps B-coordinates to standard coordinates, not automatically the reverse.
Can AI help with this MAST10007 topic?
Yes. Ask Sia to explain one step, create a fresh practice problem, or check your reasoning. Use it to learn rather than complete graded assessment, and follow University of Melbourne academic-integrity rules.
Why is a kernel basis made from parameter vectors while an image basis uses original columns?
The kernel consists of input solutions to the homogeneous equation, so its basis comes from free-variable special solutions. The image consists of actual output directions generated by the original columns, so pivot indices select those original columns rather than reduced replacements.
What is the safest check for a change-of-basis calculation?
Reconstruct the vector in standard coordinates from the source coefficients and basis, then reconstruct it again from the destination coefficients and basis. The two results must agree. This check detects both arithmetic errors and a conversion matrix used in the wrong direction.
Exam move
Use the varied 11-page chapter as a dependency map: define each object, reproduce the fresh worked example, explain both diagrams, answer the two retrieval prompts, and perform an independent check. For image, kernel, rank–nullity and change of basis, revisit the first line where your representation, dimensions, or theorem conditions diverge.
Draw a domain-to-codomain map and place the kernel on the input side and the image on the output side. For each reduction, state whether you need original pivot columns or homogeneous special solutions, then check basis sizes with rank–nullity. For coordinate conversions, write source and destination basis labels on every coefficient vector and verify the result by reconstruction.
Working through Image, Kernel, Rank–Nullity and Change of Basis in MAST10007? Sia is AskSia’s AI Mathematics tutor — ask any MAST10007 Image, Kernel, Rank–Nullity and Change of Basis question and get a clear, step-by-step explanation grounded in how MAST10007 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
