(a) Row-reduce the augmented matrix. After elimination, the third row should involve and . Identify the three qualitatively different cases for the solution set depending on the values of and .
(b) In the case where the system has a unique solution, find it (in terms of ) and verify it satisfies all three equations.
(c) In the case where the system has infinitely many solutions, write the general solution in parametric vector form. Describe the solution set geometrically.
(d) Look at the coefficient matrix in the infinite-solutions case: what is the relationship between its third row and the first two rows? Now look at the right-hand side: what relationship must satisfy relative to the right-hand sides and ? Explain why a linear dependence among the rows of the coefficient matrix forces an identical dependence on the right-hand side for the system to be consistent.
Variant 2 — All Possible Solution Geometries for a System
Consider a system of 3 equations in 3 unknowns. The solution set is determined entirely by the RREF of the augmented matrix. The rank of a matrix is the dimension of its column space: as of now, you should think of the rank as the number of pivot variables in the RREF of the matrix.
(a) There are four possible values for the rank of the coefficient matrix: . For each rank, state whether the system can be consistent for every right-hand side , or only for some , or never.
(b) For each rank, describe the solution set when the system is consistent: what geometric object is it, and how many parameters does it require?
(c) Give one explicit example of an inconsistent system for each of the three possible values of the coefficient matrix rank where inconsistency can occur.
(d) A student says: “Adding equations to a consistent system can only shrink the solution set or leave it unchanged — it can never make an originally consistent system inconsistent.” Is this true? Justify your answer.
Variant 3 — Reading a System from Its RREF
The following is the RREF of an augmented matrix:
(a) Write the general solution in parametric vector form .
(b) A second student solves the same system and arrives at using a different particular solution and , . Show that as subsets of .
(c) Write down two different systems of equations (with different coefficient matrices) that both have the RREF above. Is the RREF of a matrix unique? Is the matrix corresponding to a given RREF unique?
(d) This RREF has 3 rows. Suppose the original system had 5 equations in 4 unknowns. What must the two “extra” rows of the original augmented matrix have contributed to the RREF? What does this say about the information those equations carried?
Variant 4 — Consistency and Rank
Let .
(a) Find all for which is consistent. Describe this set geometrically and connect it to the columns of .
(b) For a consistent , describe the solution set of . Is it a subspace of ?
(c) The system is consistent if and only if the rank of equals the rank of . Verify this with one consistent and one inconsistent , and explain in one sentence why a pivot appearing in the last column is the exact obstruction.
(d) Now suppose is an arbitrary matrix with rank . Fill in the table below — for each combination, describe the solution set (or state it is empty):
consistent
inconsistent
Variant 5 — When Does a Homogeneous System Have Only the Trivial Solution?
(a) A homogeneous system has 4 equations in 3 unknowns. Can it have only the trivial solution? Can it have infinitely many? Give an explicit example of each, or explain why one is impossible.
(b) A homogeneous system has 3 equations in 4 unknowns. Can it have only the trivial solution? Can it have infinitely many? Give an explicit example of each, or explain why one is impossible.
(c) Given a homogeneous system with equations and variables. Assume it has a unique solution . What form can the RREF of the system have?
(d) A homogeneous system has rank 2. Describe all possible solution sets. Now modify the right-hand side to some : for which is the non-homogeneous system consistent? (Reason from the rank — no computation required.)
Variant 6 — Membership in a Span
Let and in .
(a) Determine whether belongs to . If it does, express it as a linear combination; if not, explain why no such combination exists.
(b) Determine whether belongs to .
(c) Find a homogeneous linear equation in the coordinates in whose solution set is exactly .
(d) Your equation in (c) shows that is the solution set of a homogeneous equation. Is this a coincidence? In general, is every span in the solution set of some homogeneous system — and is every solution set of a homogeneous system a span?
Variant 7 — Does This Set Span ?
Consider three sets of vectors in :
(a) For each set, form a matrix whose columns are the given vectors and row-reduce. Determine the rank of each matrix.
(b) Which of span ? Justify each answer from your row reduction in (a).
(c) For : the third vector is a linear combination of the first two. Express it explicitly. Does this affect the span? What does removing it do?
(d) In general, a set of vectors in can span only if . Explain why. Is sufficient? Give an example showing it is not.
Variant 8 — Span in Polynomial Space
Let , , in .
(a) Show that . Using this, explain why .
(b) A general element of has the form . Determine which of the following polynomials belong to this span: , , .
(c) Characterise as a subset of : find a single linear condition on the coefficients of that is equivalent to belonging to the span.
(d) Is ? If not, what is the “missing direction,” and what single polynomial would you add to to make the span all of ?
Variant 9 — Span is the Smallest Subspace
Let be a vector space and a nonempty set.
(a) In , let and let , . Verify that and that . Does ?
(b) Prove: if is any subspace of with , then . (Hint: a subspace is closed under taking linear combinations; use this directly.)
(c) Conclude from (b): is the smallest subspace containing , in the sense that it is contained in every subspace that contains . Write this as a precise statement.
(d) Use (c) to give a second proof that is itself a subspace: it is the intersection of all subspaces containing , and intersections of subspaces are subspaces (as shown in Week 3). (You may assume the intersection result without re-proving it.)
Variant 10 — Checking Linear Independence from the Definition
Consider the three vectors in :
(a) Set up the equation and write it as a homogeneous linear system in . Solve the system.
(b) Is the set linearly independent or dependent? If dependent, write down an explicit dependence relation and express one vector as a linear combination of the others.
(c) Does removing the redundant vector change the span? Verify directly that .
(d) Is linearly independent? If so, describe geometrically and find an equation whose solution set it is.
Variant 11 — Independence via Row Reduction
(a) Let , , in . Form the matrix and row-reduce. Is linearly independent?
(b) Explain the connection: why does a full set of pivots (one per column) imply linear independence, while a free column implies dependence?
(c) Now let , , . Row-reduce the matrix . Find an explicit dependence relation among .
(d)(Reflection.) Both sets and consist of three vectors in . One is independent, the other is not. What is the key structural difference? Could any four vectors in be linearly independent? Explain.
Variant 12 — Linear Independence in Polynomial Space
Consider the following subsets of :
(a) Check whether is linearly independent by setting and comparing coefficients of , , .
(b) Check whether is linearly independent by the same method.
(c) For the set that is linearly independent, show that it spans by expressing a general polynomial as a linear combination of its elements.
(d) For the set that is linearly dependent, find the dependence relation and identify which element is redundant. Describe geometrically (using the identification from Week 3).
Variant 13 — Dependence and Redundancy
Let be a vector space and let be a linearly dependent set.
(a) In , find a dependence relation among the vectors , , . Which of the three vectors is in the span of the other two? (There may be more than one answer.)
(b) Prove in general: if is linearly dependent, then some vector belongs to the span of the remaining vectors . (Hint: if with , solve for .)
(c) Prove: if , then .
(d) A student argues: “I can always remove vectors one by one from a finite linearly dependent set until the remainder is independent, and the span is preserved at every step.” Is this correct? What does this say about the relationship between dependent spanning sets and independent spanning sets?
Variant 14 — Verifying a Basis
(a) Verify that is a basis for by checking both conditions (independence and spanning).
(b) Verify that is a basis for .
(c) Write the vector in the basis — that is, find scalars such that . Are these scalars unique?
(d) Is a basis for ? Why or why not? What is the largest number of linearly independent vectors you can have in ?
Variant 15 — Finding a Basis for a Subspace
Let be the solution set of the homogeneous system:
(a) Row-reduce the coefficient matrix to RREF. Identify pivot and free variables, and write the general solution in parametric form.
(b) Read off the direction vectors from the parametric form and write . Verify that and each satisfy both equations.
(c) Show that is linearly independent. Conclude that is a basis for .
(d)(Reflection.) The procedure you used — row-reduce, identify free variables, write one direction vector per free variable — always produces a basis for the solution set of a homogeneous system. Why does it always produce a spanning set? Why does it always produce a linearly independent set? (You do not need a complete proof — a clear explanation suffices.)
Variant 16 — A Basis for a Polynomial Subspace
Let .
(a) Show that is a subspace of . (Check the three subspace conditions.)
(b) Write a general element of in terms of free parameters. (Write and impose the two conditions.)
(c) From (b), read off a basis for . Verify that and are linearly independent and that every element of is a linear combination of them.
(d) The full space has four “degrees of freedom.” Imposing each independent linear condition reduces the degrees of freedom by one. How many conditions did you impose, and how many basis elements did you end up with? Does this agree with the count? What would happen if you also imposed ?
Variant 17 — Unique Representations
A basis is more than a spanning set — it is a spanning set with no redundancy, which forces every vector to have a unique representation.
(a) Let be the standard basis for . Write as a linear combination of . Is there another way to do this?
(b) Now let , which spans but is not a basis. Find two different ways to write as a linear combination of vectors in . (This shows non-uniqueness for dependent spanning sets.)
(c) Prove: if is a linearly independent set, then every vector in has a unique representation as . (Hint: suppose has two representations; subtract them and use independence.)
(d) Conclude: is a basis for if and only if every vector in can be written as a linear combination of elements of in exactly one way. Explain why both directions of this “if and only if” follow from what was established in (c) and the definition of a basis.