Week 7.5 Tutorial Problems: Bases, Inverses, and Discovering the Determinant
Problem 1 — The Matrix of a Map in a Chosen Basis
For a linear map and an ordered basis of , the matrix of in , written , has as its -th column the coordinate vector — the coefficients needed to write in the basis .
Let be , and let with , .
(a) Verify that is a basis. Compute and , express each in the basis , and write down .
(b) Compare with the standard matrix of . Your should be diagonal. What is special about the vectors under that produces this?
(c) Prove the general principle behind your observation: if a basis has the property that for each (every basis vector is sent to a multiple of itself), then is the diagonal matrix with down the diagonal.
Problem 2 — Change of Basis
For a linear map and an ordered basis , the matrix of in , written , is the matrix whose -th column is (the coordinates of in the basis ). Let be the matrix whose columns are the basis vectors in standard coordinates; then converts -coordinates to standard coordinates: . Two square matrices and are called similar if for some invertible .
Take , , and let have standard matrix (that is, ).
(a) Write down and compute . Then compute .
(b) You should find . Check that this is exactly by computing and directly. Explain why similar matrices represent the same linear map in different bases, and why this makes “find a basis in which the matrix is simple” a natural goal.
(c) Prove the change-of-basis formula: if is the standard matrix of and has the basis as its columns, then .
Problem 3 — Integration and Differentiation as Inverse-ish Maps
Let be (the antiderivative vanishing at ), and let be .
(a) Using the standard bases and , write the matrix (a matrix) and the matrix (a matrix).
(b) Compute the products and . One is an identity matrix and one is not — which, and what does the non-identity product do to a polynomial? What does this reveal about as an “inverse” of ?
(c) Prove: if , then is injective and is surjective. Apply this to , and explain (using dimensions) why neither nor can be a genuine two-sided inverse of the other.
Problem 4 — Computing an Inverse by Row Reduction
For an invertible matrix , augmenting with the identity and row-reducing gives the inverse: .
Let .
(a) Apply the augmented-matrix method to find . Verify your answer by computing .
(b) Explain why this procedure works: what system is the -th column of solving, and why can all systems be solved at once? What happens to the left block if is not invertible, and how does that connect to ?
(c) For square matrices, a one-sided inverse turns out to be automatically two-sided. Prove it: if , then also (so ). (Consider what says about the kernel of .)
Problem 5 — Discovering Rotations and Reflections
A linear map is determined by where it sends the standard basis, and its matrix has columns and . We use this to find the matrices of the plane’s rotations and reflections.
(a) Let denote counterclockwise rotation by angle about the origin. Determine where and land, and write down the matrix of . (Sanity-check it against and .)
(b) Let denote reflection across the line through the origin making angle with the -axis. A unit vector pointing at angle reflects to the unit vector pointing at angle . Use this to find and , and write down the matrix of . Check it against the two familiar cases (the -axis) and (the line ).
(c) Using your matrix from (a), compute the product and simplify with the angle-addition formulas to prove . Then compute the product of two reflections : show it is a rotation, and find its angle. What does this say geometrically about performing one reflection after another?
Problem 6 — What Column Operations Do to the Determinant
You may use and its analogue.
(a) Start from , and compute how its determinant changes under each of the following, comparing to : (i) swap the two columns; (ii) multiply the first column by ; (iii) add the second column to the first.
(b) State a general rule for how each of the three operations — swapping two columns, scaling a column by , and adding a multiple of one column to another — changes the determinant. Test each rule once on a matrix of your choosing.
(c) Prove the rule you conjectured for “add a multiple of one column to another” (it is enough to treat the case).
Problem 7 — Does the Determinant Multiply? Does It Add?
Let and .
(a) Compute , , , and .
(b) Compare against , and against . Which arithmetic operation does the determinant seem to respect, and which does it not? Test the one you think it respects on a second pair of matrices of your own.
(c) Assume the rule you conjectured, , holds for all square matrices. Prove that then for every invertible , and deduce that an invertible matrix must have nonzero determinant.
Problem 8 — When Is a Determinant Zero?
The determinant is linear in each column separately and changes sign whenever two of its columns are swapped. Consider
(a) Suppose a matrix has two identical columns. Using only that swapping two columns negates the determinant, determine what its determinant must be. Confirm your conclusion on a matrix of your own with two equal columns.
(b) Compute and , and in each case write the third column as a combination of the first two. What do you conjecture about the determinant of a matrix whose columns are linearly dependent?
(c) Prove your conjecture.
Problem 9 — A Determinant Inside a Determinant
Recall the uniqueness of the determinant: it is the only function of the columns of a square matrix that is normalised (), multilinear (linear in each column separately), and skew-symmetric (it changes sign when two columns are swapped). Consequently, any function of the columns having all three properties must be the determinant.
Given an matrix , build the matrix with a in the top-left corner, zeros in the rest of the first row and first column, and in the remaining block:
(a) Compute and for (so is ), and once more for a matrix of your own. What relationship do you observe?
(b) Conjecture the general relationship between and . To prove it, regard as a function of the columns of ; the plan is to show is the determinant. Warm up with the easiest of the three properties: check that .
(c) Complete the argument: show that is multilinear and skew-symmetric in the columns of , and conclude by uniqueness that . Deduce as a corollary that placing a scalar in the corner instead of gives determinant .
Problem 10 — The Vandermonde Determinant
You may use the explicit and determinant formulas. For parameters , consider
(a) Compute and , and factor each result completely.
(b) Your factored forms are built out of differences of the parameters. Notice that vanishes whenever two of are equal — explain this directly from the matrix (what happens to its columns when, say, ?). Use this to predict which linear factors must contain, and hence its shape up to a constant.
(c) Prove that equals exactly — including that the constant is . (You might compare the degree of the determinant, as a polynomial in , with the degree of the product of factors from (b).) Finally, can you guess a general conjecture for the Vandermonde determinant, whose -th column is ?
Problem 11 — Which Quantities Survive a Change of Basis?
Two square matrices and are similar if for some invertible — they are the same linear map written in two different bases. You may use that the determinant is multiplicative: . Let
(a) Compute . Then, for both and , compute four numbers: the determinant, the trace (the sum of the diagonal entries), the sum of all four entries, and the top-left entry. Which of the four agree between and , and which do not?
(b) From your table, conjecture which of these quantities are similarity invariants — the same for every matrix similar to . For the determinant, pin down exactly why forces .
(c) Prove that the determinant is a similarity invariant. Conclude that the determinant of a linear map is well defined, independent of the basis used to represent . (One of the other quantities you found is also invariant — you need not prove that here.)
Problem 12 — Determinants Forced by an Equation
A square matrix is idempotent if (these are the projections) and nilpotent if for some . You may use that the determinant is multiplicative: . Consider the projection and the nilpotent matrix .
(a) Compute and , and check that and .
(b) Using multiplicativity, relate to , and to . What values are then forced for the determinant of an idempotent matrix, and for that of a nilpotent matrix? Conjecture each answer.
(c) Prove your conjectures: the determinant of any idempotent matrix is or , and the determinant of any nilpotent matrix is . Then, for a further challenge: show that for any idempotent matrix , the matrix satisfies , and deduce that .