MAT 242 Chap.7 Bases, Dimension and Coordinates
Bases, Dimension and Coordinates
A basis does two jobs
A basis for a subspace spans it and is independent, so it generates every vector with nothing wasted. Subspaces have many bases, but they all contain the same number of vectors, and that number is the dimension. In the plane the possibilities are the origin alone, lines through the origin and the whole plane; no set of more than n vectors in Rⁿ can be independent.
A basis is found from a spanning set by row reducing and keeping the original vectors in the pivot columns.
Coordinates
Because a basis is independent, each vector in the subspace has exactly one list of weights, and that list, taken in the order of an ordered basis, is its coordinate vector.
When the basis vectors form the columns of an invertible matrix, coordinates come from multiplying the vector by that matrix's inverse, and the vector is rebuilt by multiplying the coordinates by the matrix itself.
Changing basis
The same vector has different coordinates in different bases.
Rebuilding the vector from one basis and reading it in another gives the transition matrix, the inverse of the second basis matrix times the first.
The order is where marks are lost, so the paid chapter works a transition matrix in the plane, checks it through the vector itself, and drills the direction of conversion.
The written homework and Test 2 both ask for coordinates and conversions, and the course expects the matrices you enter into the calculator to be written down alongside the result.
What Test 2 asks
The Test 2 list includes finding the dimension of a subspace from a spanning set, finding coordinates relative to a basis, converting coordinates between two bases and finding the change-of-basis matrix.
Sample solutions give heavily reduced credit when the conversion matrix is multiplied in the wrong order, so the order check on the paid page is worth memorising.
When a basis happens to be orthogonal, a quicker coordinate method becomes available later in the course, but it must never be used on an ordinary basis.
Entering a basis in the online homework
The course notes show the format the online homework expects: each basis vector written in square brackets with its entries separated by commas, and the vectors separated by commas.
Written work is different: there a basis is presented as a set of column vectors, and the course accepts factored forms such as a fraction multiplying a vector. Remember that a subspace has many correct bases, so a basis that differs from a classmate's is not automatically wrong; check it by confirming independence and the right number of vectors.
What this chapter covers
- 01
Basis and dimension
- 02
Subspaces of the plane by dimension
- 03
Finding a basis from a spanning set
- 04
Coordinate vectors for an ordered basis
- 05
The change-of-basis matrix
Worked example · free
Coordinates in a two-vector basis
- 2Solve c₁ + c₂ = 4 and c₁ − c₂ = 1; adding gives 2c₁ = 5.
- 1So c₁ = 5/2 and c₂ = 3/2.
- 1Rebuild: (5/2)(1, 1) + (3/2)(1, −1) = (4, 1).
Key terms
- Basis
- A basis is a linearly independent set that spans a subspace.
- Dimension
- Dimension is the number of vectors in any basis of a subspace.
- Transition Matrix
- A transition matrix converts coordinates relative to one ordered basis into coordinates relative to another.
Bases, Dimension and Coordinates FAQ
Why does the order of basis vectors matter for coordinates?
Coordinates are listed in the same order as the basis vectors. Reordering the basis permutes the coordinate entries, so the course writes ordered bases with parentheses rather than set braces.
Which matrix converts B-coordinates into C-coordinates?
Multiply the B-coordinates by the matrix of B first, which rebuilds the vector, then by the inverse of the matrix of C. The transition matrix is C̃ inverse times B̃, in that order.
How do I find the dimension of a subspace from a spanning set?
Put the spanning vectors in as columns, row reduce, and count the pivots. That count is the dimension, and the original vectors in the pivot columns form a basis.
Exam move
For every coordinate or change-of-basis answer, rebuild the actual vector in both bases. It takes a minute and is the most reliable way to catch an inverse placed on the wrong side. Practise each change-of-basis question twice: once with the formula and once by rebuilding the vector, and compare the two answers.
Working through Bases, Dimension and Coordinates in MAT 242? Sia is AskSia’s AI Mathematics tutor — ask any MAT 242 Bases, Dimension and Coordinates question and get a clear, step-by-step explanation grounded in how MAT 242 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
