(a) Compute and . Compute and . Is commutative? Associative?
(b) Find the identity element for — the element with for all . This plays the role of the zero vector.
(c) Find the additive inverse of under — the element with . (Your is from part (b).)
(d) Check axiom (S1): is for all ? Check axiom (D2): is for all and ?
(e)(Challenge.) The map defined by converts and into the standard operations on : verify that and . What does this say about the relationship between and the standard ?
Variant 2 — Two Vector Spaces in Disguise
Let be the set of polynomials of degree at most 2, with the usual operations and . Define a map by
(a) Compute and . What vectors in do you get?
(b) Let and . First compute in , then apply to get . Separately, compute in . Are the two results equal?
(c) Now compute and . Are these equal?
(d) Based on (b) and (c), write down two equations that appears to satisfy — one involving addition, one involving scalar multiplication. Do you expect these to hold for every and every ? Try to explain why, without computing case by case.
(e) Is invertible? Given any triple , can you always recover a unique polynomial in ? What does this say about how and relate to each other as vector spaces — are they genuinely different, or are they somehow “the same”?
Variant 3 — A Broken Operation on
Define operations on by:
(a) Compute and . Is commutative?
(b) Find an element satisfying for all . (Compute and see what must be.)
(c) Does the vector have an additive inverse under ? That is, can you find with ? What goes wrong?
(d) Check axiom (D1): is ? Test with , , .
(e) Make a precise list: which of the eight vector space axioms hold for , and which fail? For each failure, give a specific counterexample.
Variant 4 — The Complex Numbers as a Real Vector Space
The complex numbers form a set. Define two operations with real scalars only:
(a) Compute and . What are the results?
(b) What is the zero vector in ? What is the additive inverse of ?
(c) Check axiom (D2): . Verify with , , . (Does the result hold? Can you see why it holds in general?)
(d) Let be the set of complex numbers with zero imaginary part — the “real axis.” Check all three subspace conditions: is ? Is closed under addition? Under scalar multiplication?
(e)(Discovery.) Define by (multiply by ). This is not scalar multiplication since . But check: is ? Is for ? Compute and . Sketch where sends the vectors and in the plane . What geometric transformation is ?
Variant 5 — Matrices as Vectors
Let be the set of all real matrices, with entry-by-entry addition and scalar multiplication:
(a) Let and . Compute and .
(b) What is the zero vector in ? What is the additive inverse of ?
(c) The trace of a matrix is the sum of its diagonal entries: . Let . Check all three subspace conditions.
(d) Let be the set of invertible matrices — those for which . Is a subspace of ? If not, which condition fails first?
(e)(Reflection.) A matrix is determined by its four entries . How do the operations on compare to the standard operations on ? Can you write down a map that converts matrix addition into vector addition? Does it handle scalar multiplication the same way?
Variant 6 — From Geometry Back to Equations
In lecture we went from an equation to a geometric picture. This problem goes the other way: start from a geometric object and find the equations that cut it out.
(a) The vector spans a line through the origin in . Find an equation whose solution set is exactly this line.
(b) Is your equation from (a) the only one? Find a second equation with the same solution set. What is the relationship between all valid equations?
(c) Now work in . You want a plane through the origin containing both and . A plane through the origin has an equation . Write the conditions that and both satisfy this equation. Solve for and write down the equation of the plane.
(d) Re-examine part (a) from the row picture. The solution set of consists of all vectors in perpendicular to the row vector . Verify: is perpendicular to the you found in (a)?
(e)(Dimension counting.) A single homogeneous equation in cuts out a line (dimension 1 = 2 − 1). In , a single equation cuts out a plane (dimension 2 = 3 − 1). Suppose you want to describe a line through the origin in using equations. How many equations do you need? What about a line through the origin in ?
Variant 7 — Solution Set of a Homogeneous Equation in
Consider the equation
(a) Find two solutions that are not scalar multiples of each other.
(b) Set and (two free parameters) and express in terms of and . Write the solution set in the form for specific vectors .
(c) What kind of geometric object is the solution set? Does it pass through the origin? What does it mean, geometrically, that the solution set requires two parameters?
(d) Now consider . Find one particular solution . Write the general solution in the form , where and are from (b). How does this solution set relate geometrically to the homogeneous one?
(e)(Reflection.) Each equation in has a solution set that is a “flat object” through the origin. In , a single equation gives a line (1-dimensional). In , a single equation gives a plane (2-dimensional). What pattern do you see? What would the solution set of a single homogeneous equation in look like?
Variant 8 — Two Homogeneous Equations in
Consider the system:
(a) Use arithmetic operations with the equations to determine all solutions.
(b) Write the solution set in the form for a specific direction vector . What geometric shape is this?
(c) Interpret this geometrically: the solution set of each equation separately is a plane through the origin. Their intersection is the solution set of the system. What kind of object is the intersection of two distinct planes through the origin in ?
(d) Verify directly that if and both satisfy both equations in the system, then so does and .
(e)(Challenge.) Suppose we add a third equation to the system. Does the solution set change? (Hint: check whether your from (b) satisfies the new equation.) In general, what are the possible shapes for the solution set of a homogeneous system of equations in ?
Variant 9 — The Column Picture
Consider the system:
(a) Rewrite the system in the form , identifying the column vectors and the right-hand side vector .
(b) Solve the system by substitution or elimination to find and .
(c) Verify your solution: compute directly and confirm it equals . Sketch the vectors , , and in , showing how the first two add up to the third.
(d) Now change the right-hand side: replace with to get the homogeneous system . Solve it. What is the only solution?
(e)(Reflection.) The column picture reframes “solving a system” as “can be expressed as a combination of the columns?” Choose a pair of column vectors and . For which right-hand sides does have a solution? Describe this geometrically.
Consider the system with one equation in three unknowns:
(a) Find one particular solution satisfying the equation. (There are many choices — pick the simplest.)
(b) Write the general solution of the associated homogeneous equation in the form for two free parameters .
(c) Verify that satisfies the original equation for every .
(d) Show that every solution of can be written in this form. (Hint: if is any solution, show that satisfies the homogeneous equation.)
(e)(Reflection.) A student argues: “I found a different particular solution . My general solution looks different from yours. We can’t both be right.” Explain why you are both right — the set of all solutions and are the same set.
Variant 11 — Which Subsets of Are Subspaces?
A subset of a vector space is a subspace if (i) , (ii) is closed under addition, and (iii) is closed under scalar multiplication.
Consider the following subsets of :
(a) Which of contain the origin ? Eliminate any that do not — they cannot be subspaces.
(b) For those containing the origin: check whether they are closed under addition. For any that fail, exhibit two specific vectors in the set whose sum is not in the set.
(c) For those passing (b): check whether they are closed under scalar multiplication. For any that fail, exhibit a specific scalar and vector whose product is not in the set.
(d) Conclude: which of is a subspace? Justify each answer with a complete check or a specific counterexample.
(e)(Pattern recognition.) Describe the geometric shape of each . Based on your findings, formulate a conjecture: which geometric shapes in are subspaces?
Variant 12 — Subspaces of
Consider the following subsets of :
(a) For each of : does it contain the zero vector? Show your work.
(b) For each subset containing : check closure under addition. For any that fail, give a concrete counterexample.
(c) For each subset passing (b): check closure under scalar multiplication. For any that fail, give a concrete counterexample.
(d) Conclude which of are subspaces. For each subspace, describe it geometrically (line, plane, etc.).
(e)(Extension.) The set is the intersection of two subspaces. Describe it geometrically. Is it a subspace? Check the three conditions.
Variant 13 — Subspaces of Polynomial Space
Let be the vector space of all polynomials with real coefficients.
(a) Let . Show that is a subspace of by verifying all three conditions. (Hint: if and , what is ?)
(b) Let . Does contain the zero polynomial? Is closed under addition? Give a specific example to show which condition first fails.
(c) Let (polynomials of degree at most 3). Verify all three subspace conditions. Is the sum of two polynomials of degree again of degree ?
(d) Let . Determine whether is a subspace. (Hint: check whether , whether the sum of two elements of is in , and whether scalar multiples stay in .)
(e)(Reflection.) Of the subspaces you found, is defined by a constraint on the value at a point, and is defined by a bound on the degree. Find a polynomial space that satisfies both constraints simultaneously — one that is a subspace of and of .
Variant 14 — The Intersection and Union of Subspaces
Let and be subspaces of a vector space .
(a) Take , (the -plane), and (the -plane). Describe as a set. What geometric object is it?
(b) Verify that is a subspace of by checking all three subspace conditions directly.
(c) Prove in general: if and are any two subspaces of a vector space , then is also a subspace. (Hint: for each of the three conditions, use that and are each subspaces.)
(d) Now consider the union of the same and from (a). Find a vector in and a vector in whose sum is in neither nor . Conclude that is not a subspace in general.
(e)(Reflection.) So intersection preserves the subspace property, but union does not. What would the “smallest subspace containing both and ” need to include? (You do not need to prove anything — just describe what vectors must be in this set and why.)
Variant 15 — Scalar Zero and Vector Zero
Two facts that look obvious turn out to require proof: that (the scalar kills any vector), and that (any scalar kills the zero vector). Neither follows directly from the axioms — they must be deduced.
(a) In , compute directly from the definition of scalar multiplication. What do you get? Is the result surprising?
(b) Now prove in any vector space. The argument starts by writing in and applying axiom (D2):
You now have . How do you conclude ?
(c) In , compute directly. What do you get?
(d) Prove in any vector space. Use the same strategy as (b), but write using axioms of a vector space. Complete the argument.
(e)(Reflection.) Both proofs use the same trick: write something as “itself plus itself” and then cancel. Identify which axiom does the cancellation in each case. Why is it not enough to say “because times anything is “? What would be circular about that reasoning in this context?
Variant 16 — The Additive Inverse and
Axiom (A4) guarantees that every vector has an additive inverse — a vector with . This problem proves two things: that is exactly that inverse, and that it is the only one.
(a) In , compute from the definition of scalar multiplication. What is the result? Does it match your expectation for ?
(b) Prove that satisfies in any vector space. (Hint: write using (S1), then compute using (D2). You will need , which was proved in Variant 15.)
(c) Prove that additive inverses are unique: if and , then . (Hint: add to both sides of , and use associativity and the first equation.)
(d) Using (b) and (c) together: why can we now unambiguously write and know that ?
(e)(Extension.) What is ? Give two ways to see this: one using the definition of additive inverse, and one using the result from (b) applied twice.
Variant 17 — Why Homogeneous Solution Sets Are Subspaces
Recall from the lectures that the sum of two solutions to a homogeneous equation is again a solution, because . This problem turns that observation into a complete subspace proof — and pinpoints exactly where the argument breaks for non-homogeneous equations.
(a) Let be the solution set of the system:
Check that . Find one non-zero solution.
(b) Suppose and both satisfy both equations. Show step by step that also satisfies both equations. Write out the algebra for each equation separately.
(c) Suppose satisfies both equations and . Show that satisfies both equations.
(d) Using (a), (b), (c): conclude that is a subspace of .
(e)(The non-homogeneous case.) Now let be the solution set of the same system but with right-hand sides changed to :
Does contain the zero vector? Is closed under addition? Find a specific pair of solutions in whose sum is not in . Where does your proof from (b) break down?