Variant 1 — Why More Than Vectors Cannot Be Independent
Let be a basis of , and let
(a) Write each of as a linear combination of and .
(b) Arrange the coordinate pairs from (a) as columns of a matrix . What are the dimensions of , and what does this imply about the homogeneous system ?
(c) Prove that is linearly dependent.
(d) Prove the general statement: if and , then are linearly dependent. Note that the need not themselves be linearly independent — make sure your proof does not use their independence.
Variant 2 — Dimension of Familiar Spaces
(a) Find a basis and state the dimension for each of the following subspaces. (i) . (ii) .
(b) Let be the vector space of symmetric real matrices (symmetric matrices have equal elements in elements symmetric with respect to the main top-left to bottom-right diagonal). Find a basis and state its dimension.
(c) Prove: if and is a subspace with , then .
(d) Two of the spaces in (a)–(b) have the same dimension. Does this make them “the same” as vector spaces? What additional data would be needed to declare them genuinely indistinguishable?
Variant 3 — The Steinitz Exchange in Action
Start with the standard basis of and the linearly independent set
(a) Express in terms of . Choose a basis vector whose coefficient is non-zero, replace it with to form , and show that spans .
(b) Express in terms of . At least one coefficient of a remaining -vector must be non-zero — explain why, using linear independence of . Perform the exchange to get and verify it spans .
(c) Perform the third exchange with to arrive at , and verify is a basis of .
(d) Suppose we now try to run a fourth exchange with some . Where exactly does the procedure break down, and why? Use this to explain the proof that a linearly independent set can have no more elements than any spanning set.
Variant 4 — The Dimension Shortcut
(a) In , let . Show that is linearly independent, and conclude — without checking span directly — that is a basis of .
(b) Let . Find . Then decide, with minimal computation, whether is a basis of .
(c) A student claims: “The set has 3 vectors in , so by the dimension shortcut it is a basis.” Identify the precise logical error, and find an explicit dependence relation confirming the set is not a basis.
(d) Prove both directions of the dimension shortcut: (i) if and spans , then it is linearly independent; (ii) if and is linearly independent, then it spans .
Variant 5 — Dimension and Subspaces of
(a) Let and . Find a basis and dimension for each.
(b) Find ; give a basis and state its dimension.
(c) List all possible dimensions a subspace of can have. For each, give a concrete example and a geometric description. What is the only -dimensional subspace of , and why?
(d) Determine whether and verify the formula . This formula is reminiscent of inclusion-exclusion from set theory. Is this similarity a coincidence?
Variant 6 — Computing Coordinate Vectors
Let be a basis of .
(a) Find for , , and .
(b) Let be a basis of . Find for and for .
(c) Find and , where is the standard basis.
(d) The polynomial is a single mathematical object, yet its coordinate vector changes with the basis. What exactly changes when you switch from to , and what does not? Which is the “real” object: the polynomial or the coordinate vector?
Variant 7 — The Coordinate Map Is Linear
Fix a basis of a -dimensional vector space . Suppose and .
(a) Compute and without knowing or the basis vectors explicitly.
(b) Prove: for any and scalar ,
(c) Prove that the coordinate map defined by is a bijection (one-to-one function).
(d) A map that is both linear and bijective is called a linear isomorphism; two vector spaces are isomorphic if one exists between them. Determine, without computation, whether and whether . What invariant distinguishes non-isomorphic finite-dimensional spaces?
Variant 8 — Finding Coordinates via a Linear System
Let be the basis of from Lecture 7.
(a) Find for and .
(b) Find a formula for in terms of the entries of a general , and express the result as a matrix multiplication .
(c) The standard basis satisfies for all , but does not. Identify why: what is when ?
(d) The problem of finding is a problem of the form , which we have been solving since Week 2. Identify precisely what plays the role of and here, and explain why “find the coordinates of in basis ” and “express as a linear combination of the columns of ” are two phrasings of the same problem.
Variant 9 — Coordinates Depend on the Basis; the Vector Does Not
Work in with the bases and .
(a) Verify that is a basis of . Find and .
(b) Find for a general polynomial , expressing each -coordinate as a function of .
(c) Your answer to (b) converts into by a linear formula. Find the matrix such that for all , and verify with .
(d) The matrix is called a change-of-basis matrix. Prove that is invertible, and explain what represents. Then give a general argument: why must the change-of-basis matrix between any two bases of the same finite-dimensional space always be invertible?
Variant 10 — Linear Independence via Coordinates
Let be a vector space with basis , and let .
(a) Prove: is linearly independent in if and only if is linearly independent in .
(b) Let be a basis of . Define , , . Determine whether is linearly independent without computing the vectors explicitly.
(c) In some 3-dimensional space with basis , three vectors have coordinate vectors , , . Find , a basis for this span, and express any dependent vector as a linear combination of the basis — entirely without knowing the actual vectors in .
(d) Prove that the coordinate map also preserves spans: . Conclude that studying subspaces of any -dimensional space is equivalent to studying subspaces of .
Variant 11 — Coordinates as a Measuring Device
Let be a -dimensional vector space with basis , and suppose and .
(a) Determine , , and without knowing or the basis vectors.
(b) Let . Describe the image as a subset of , find a basis for it, and state its dimension.
(c) Prove that is a subspace of , and conclude that .
(d) The previous parts show that any question about the linear structure of translates, via , into a question about . What does this say about the relationship between , , and as objects of linear algebra? What is lost or gained by passing to coordinates?
Variant 12 — Reducing a Spanning Set to a Basis
Consider the five vectors in :
(a) Row-reduce and identify the pivot columns.
(b) State a basis for , its dimension, and whether it equals .
(c) Express each non-pivot vector as a linear combination of the pivot-column vectors.
(d) Prove that the pivot-column method always works: the pivot columns of a matrix are linearly independent and span the same space as all the columns of .
Variant 13 — Dependence Relations Form a Subspace
Let , , , in .
(a) Find a basis for consisting of polynomials from the given set, and state . Express each non-basis polynomial as a linear combination of your basis.
(b) A dependence relation among is a tuple satisfying . Find all dependence relations — that is, solve the homogeneous system obtained by comparing coefficients of — and write down a basis for the solution set.
(c) The solution set in (b) is a subspace . State . What is the relationship between , , and the number of polynomials ?
(d) Prove the general statement: if are vectors in a vector space and , then . Explain each term: what does count, what does count, and why do they sum to ?
Variant 14 — Extending a Linearly Independent Set to a Basis
Let and in .
(a) Verify that is linearly independent and state .
(b) Extend to a basis of .
(c) Find a second extension, with different completing vectors than in (b), and verify it is also a basis.
(d) Prove: any linearly independent set in a finite-dimensional vector space can be extended to a basis.
Variant 15 — Bases for Intersections and Sums
Let and in .
(a) Verify that the two given spanning sets are linearly independent and state and .
(b) Find ; give a basis and state its dimension.
(c) Find a basis for and verify:
(d) Explain what the term in the formula is “correcting for,” and identify the analogous term in the set-theoretic inclusion-exclusion formula. Is the analogy a coincidence?
Variant 16 — Two Characterisations of a Basis
(a) In , let . Find all single vectors whose removal from preserves .
(b) Let in . Find two distinct vectors, each of which extends to a basis.
(c) Prove: is a basis for if and only if is a minimal spanning set — that is, spans but no proper subset of does.
(d) Prove: is a basis for if and only if is a maximal linearly independent set — that is, is linearly independent but is dependent for every .