Week 6 Tutorial Problems: Practicing for the midterm
Problem 1 — Linear Independence via Coordinates
Let be a -dimensional vector space with basis , and let with
(a) Determine whether is linearly independent by working only with the coordinate vectors. If dependent, find the dependence relation among .
(b) Find and a subset of that forms a basis for this span — without knowing the actual vectors in .
(c) Prove: is linearly independent in if and only if is linearly independent in .
Problem 2 — Reducing a Spanning Set
Consider the following five polynomials in :
(a) Write the coordinate vector of each in the standard basis , form the matrix with these as columns, and row-reduce. Identify the pivot columns and extract a basis for .
(b) Express each non-basis polynomial as a linear combination of your basis. Identify the dependence relation for and verify it holds as a polynomial identity.
(c) Prove: the pivot columns of a matrix are linearly independent.
Problem 3 — Extending to a Basis
Let and in .
(a) Verify that is linearly independent. Find a vector such that is a basis of , and verify your answer.
(b) Is unique? Find a second vector that also extends to a basis of . Characterize all vectors that cannot extend the set (i.e., all vectors such that is not a basis).
(c) Prove: any linearly independent set in a finite-dimensional vector space can be extended to a basis.
Problem 4 — Counting Degrees of Freedom
Let .
(a) Write , impose the two conditions, and find a basis for .
(b) State . You imposed two conditions on a -dimensional space and reduced the dimension by . Under what circumstances would two written conditions only reduce the dimension by instead of ? Give an explicit example of two conditions on that together only reduce the dimension by .
(c) Prove: if is an -dimensional vector space and is a nonzero linear functional (meaning satisfies and ), then .
Problem 5 — Proper Subspaces and Dimension
Let in .
(a) Find . Then determine whether belongs to .
(b) The sum of two subspaces is defined as . If , find . Is ?
(c) Prove: if is a proper subspace of a finite-dimensional vector space (in other words, ), then .
Problem 6 — A Comprehensive Basis Problem
Let and .
(a) Find and determine whether .
(b) Find a polynomial such that is a basis of . Is unique? Describe all polynomials that work as .
(c) Let be the basis from (b). Prove that every polynomial can be written as in exactly one way.
Problem 7 — How Constrained Is a Linear Map?
You want to define a linear map .
(a) You set . Using only the definition of linearity, compute . Describe the set geometrically.
(b) A classmate claims: “I can independently set , regardless of your choice above.” Is the classmate right?
You now also set . After specifying on both vectors, is there any vector in whose image you can still freely choose?
(c) Prove: if is linear and in , then
What does this tell you about the freedom you have when specifying values of ?
Problem 8 — A Linear Map Determined by Its Values on a Basis
Let be a -dimensional vector space with ordered basis , and let be a linear map with
(a) Compute without knowing what the basis vectors look like.
(b) For a general with , express as a matrix-vector product . What is ?
(c) Prove: a linear map is completely determined by its values on any basis of . That is, if is linear and for all , then for every .
Problem 9 — The Kernel
Let be defined by where
(a) Row-reduce and find all solutions to . Express as a span of explicit vectors.
(b) Find a basis for . State (the nullity) and (the rank), and verify the rank-nullity theorem.
(c) Prove: for any linear transformation , is a subspace of the domain.
Problem 10 — Polynomial Interpolation
Define by
(a) Find the matrix of with respect to the standard basis of and the standard basis of . Compute .
(b) Row-reduce . What is ? Use rank-nullity to determine and identify .
(c) For any , construct an explicit polynomial with , , . What can you conclude about ?
Problem 11 — Injectivity and the Kernel
A map is injective if implies : distinct inputs always produce distinct outputs.
(a) Let have matrix . Find and determine whether is injective.
(b) Let have matrix . Find and determine whether is injective.
(c) Prove: a linear map is injective if and only if .
Problem 12 — Consequences of Rank-Nullity
A map is injective if distinct inputs have distinct outputs (), and surjective if every element of is an output ().
(a) Let be any linear map. Using only the rank-nullity theorem, explain why cannot be surjective.
(b) Let be any linear map. Using only the rank-nullity theorem, explain why cannot be injective.
(c) Prove: if is a linear map with , then is injective if and only if is surjective.
Problem 13 — Composition of Linear Maps
Let and have matrices
(a) Compute the matrix of and the matrix of .
Hint: to find the matrix of a linear map, apply it to each standard basis vector and place the results as columns. For , apply first, then .
(b) Find . Using rank-nullity, explain why cannot be the identity map on .
(c) Prove: if and are linear, then is linear.
Problem 14 — The Trace as a Linear Map
Let denote the space of real matrices, and define by .
(a) Show that is linear. Write a general element of in terms of free parameters.
(b) State and verify the rank-nullity theorem (recall ).
(c) Prove that the three matrices
form a basis for .
Problem 15 — Composing Transformations and Discovering Matrix Multiplication
Let be rotation by counterclockwise and be reflection across the -axis:
(a) Find the matrices of and of by computing the images of and under each map.
(b) Find the matrix of by computing and directly. The columns of are and : apply to each of them and compare the results to the columns of . What pattern do you see?
Now find the matrix of the same way. Is ?
(c) Prove the pattern from (b) in general: if and have matrices and respectively (standard bases), then the -th column of the matrix of equals applied to the -th column of .
(This is exactly what matrix multiplication will be defined to compute: the product is the matrix whose -th column is applied to the -th column of .)
Problem 16 — A Map from an Infinite-Dimensional Space
Let denote the space of all polynomials (of any degree), and define by
(a) Compute , , , . Verify that , , and all lie in .
(b) Show is linear. Consider the polynomials for ; argue that they are all in and are linearly independent (look at their degrees). What does this tell you about ? Can you apply the rank-nullity theorem?
(c) Prove : given any , construct an explicit polynomial with and .