MAST10007 Chap.7 Basis, Dimension, Coordinates and Fundamental Spaces
Basis, Dimension, Coordinates and Fundamental Spaces
Chapter 7 develops basis, dimension, coordinates and fundamental spaces for University of Melbourne MAST10007. A basis is a coordinate system: it spans the space without redundancy, so every vector has one coordinate list. Row reduction reveals pivot locations and null-space equations, but row operations change the actual columns.
Therefore a column-space basis must use pivot columns from the original matrix, whereas nonzero rows of an echelon form can form a row-space basis. The four fundamental spaces fit together through rank and nullity.
It pairs exact definitions with a new completed example, two concept diagrams, diagnostic contrasts and answered retrieval prompts.
The same reduction supports several outputs, but each output must be taken from the correct place. Pivot indices select original columns for a column-space basis; nonzero echelon rows represent the row space; free-variable special solutions form a null-space basis.
Coordinates are ordered coefficients relative to a declared basis and must be checked by reconstruction in the ambient space. Dimension then acts as a consistency audit across these objects. The goal is not simply to count pivots, but to connect each count and vector list to the space it actually belongs to.
What this chapter covers
- 01
Basis as span plus independence
- 02
Dimension
- 03
Coordinates relative to a basis
- 04
Extracting a column-space basis
- 05
Row-space basis
- 06
Null-space basis
- 07
Left null space
- 08
Four fundamental spaces
- 09
Pivot bookkeeping
- 10
Dimension consistency checks
- 11
Original-column versus echelon-row bookkeeping
- 12
Ordered-coordinate reconstruction checks
- 13
Connecting rank and nullity to basis sizes
Basis, Dimension, Coordinates and Fundamental Spaces worked example
- stepSeek coefficients a,b with a(1,1)+b(1,-1)=(5,1).
- stepThe component equations are a+b=5 and a-b=1.
- stepAdding gives 2a=6, so a=3; then b=2.
- stepThus [x]_B=(3,2). Reconstructing 3(1,1)+2(1,-1)=(5,1) checks both coordinate direction and arithmetic.
Key terms
- Basis as span plus independence
- A basis combines two properties: spanning ensures every vector can be expressed, and independence makes that expression unique. Omitting either property leaves a generating list or an independent list, not a coordinate system.
- Dimension
- Dimension is the number of vectors in any basis of a finite-dimensional space. All bases have the same size. This invariant turns pivot counts into geometric information and rules out impossible proposed bases immediately.
- Coordinates relative to a basis
- For an ordered basis B, [v]_B is the list of coefficients in v=c1b1+⋯+cnbn. Reordering the basis reorders or changes coordinates even though v itself is unchanged. Reconstructing v is the decisive direction check.
- Extracting a column-space basis
- A basis for Col(A) uses the columns of the original A whose indices are pivot columns after reduction. The reduced matrix reveals which indices matter but generally has a different column space, so its pivot columns are not the requested original vectors.
- Row-space basis
- The nonzero rows of an echelon form provide a basis for Row(A). Row operations preserve row space because each new row is a combination of old rows and the operations are reversible.
- Null-space basis
- Solve Ax=0, introduce one parameter per free variable, and separate coefficients of those parameters. The resulting special-solution vectors form a basis for Nul(A); their number is the nullity.
- Pivot bookkeeping
- Reduction identifies pivot positions, but the requested basis determines where the vectors come from. Use original pivot columns for the column space, nonzero echelon rows for the row space, and special solutions of the homogeneous system for the null space.
- Four fundamental spaces
- The column space and null space are associated with the matrix, while the row space and left null space connect to its transpose. Rank controls the dimensions of the row and column spaces, and nullity accounts for the remaining domain directions.
Basis, Dimension, Coordinates and Fundamental Spaces FAQ
How do I check basis as span plus independence?
A basis combines two properties: spanning ensures every vector can be expressed, and independence makes that expression unique. Omitting either property leaves a generating list or an independent list, not a coordinate system. Three, by nullity n-r=5-2.
What is the main trap in basis, dimension, coordinates and fundamental spaces?
Do not take pivot columns from the reduced matrix when asked for a basis of the original column space. Coordinates are coefficients relative to an ordered basis, not the vector's standard entries.
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 can I use echelon rows but not echelon columns for the original spaces?
Row operations preserve the row space because each new row is a reversible linear combination of old rows. They generally change the column vectors themselves, so reduction supplies pivot indices for the original column space rather than replacement column vectors.
How do coordinates differ from the vector's standard entries?
Coordinates record coefficients relative to a declared ordered basis. They equal standard entries only when that basis is the standard basis in the same order. Reconstructing the vector from its basis vectors is the most direct way to confirm the coordinate direction and ordering.
Exam move
Use the varied 10-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 basis, dimension, coordinates and fundamental spaces, revisit the first line where your representation, dimensions, or theorem conditions diverge.
Make a source table for every requested basis: original pivot columns for the column space, echelon rows for the row space, and parameter vectors for the null space. Label the ambient space and expected dimension before listing vectors. For coordinate questions, preserve basis order and reconstruct the vector. Use rank and nullity as final consistency checks rather than as substitutes for the basis argument.
Working through Basis, Dimension, Coordinates and Fundamental Spaces in MAST10007? Sia is AskSia’s AI Mathematics tutor — ask any MAST10007 Basis, Dimension, Coordinates and Fundamental Spaces 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.
