Past Exam study guide

Exam 2 (Wed, April 20) study guide

Link to Lecture videos or lecture notes from lecture 9a to 14a
See pictures of week 14 board work (Canvas login required), on Monday some questions from the list below were done on the board

Review questions for extra credit 4

Computation, definitions, and arguments to practice:

From 9a Vector geometry
- Use drawing to explain the geometric meaning of a scalar multiplication. See the end of the PDF notes.
- Use drawing to explain the geometric meaning of the sum of two vectors. See the end of the PDF notes.
From 9b Vector geometry with matrices
- Given a specific matrix A and the linear transformation T_A, what is the domain of T_A? (Answer: vectors with height k, where k is number of columns of A)
- What is the target of T_A? (Answer: vectors with height k, where k is number of rows of A)
From 10a Linear transformations
- Def: What does it mean to say that a function T preserves linear combination?
- Given a mystery linear transformation T, use the fact that T preserves linear combination to compute where T sends a vector. See Exercise 2.
- If T is a linear transformation, must T send a zero vector to a zero vector? (Answer: Yes, it’s one of the properties of a linear transformation)
- Is 2D translation to the right by 10 a linear transformation? (Answer: No, because if point at the origin is translated to the right by 10 then the result is no longer the origin)
- If a function F sends a zero vector to a zero vector, must F be a linear transformation? (Answer: No. Counterexample: In Exercise 1, the function F maps the zero vector to the zero vector, but F does not preserve adddition, so F is not a linear transformation)
From 10b Linear transformations, part b
- You should know how to write down the standard basis vectors. In later lectures, we learn that the set of standard basis vectors forms a basis for R^n.
From 11a Subspaces
- Def: What does it mean to say that W is a subspace of {height-n vectors}?
- Def: What does it mean to say that a subset W is closed under addition?
- Def: What does it mean to say that a subset W is closed under scalar multiplication?
- Def: What is the kernel of a matrix M? Is it always a subspace? (Yes, the kernel of a matrix is always a subspace)
- Def: What is the image of a matrix M? Is it always a subspace? (Yes, the image of a matrix is always a subspace)
- Def: What is the 1/2-eigenspace of a matrix M? Is it always a subspace? (Yes, the 1/2-eigenspace of a matrix is always a subspace. If 1/2 is not an eigenvalue of M, then the 1/2-eigenspace consists of only the zero vector.)
From 11b Subspaces, how to write proofs
- Given a subset W of vectors (with a simple description similar to the lecture examples), show that W is not empty. (Produce one vector which is in W)
- Given a subset W of vectors (with a simple description similar to the lecture examples), show that W is closed under addition.
- Given a subset W of vectors (with a simple description similar to the lecture examples), show that W is closed under scalar multiplication.
- Given a subset W of vectors,
  - show that W is not closed under addition. (You need to give one specific counterexample.)
  - show that W is not closed under scalar multiplication. (You need to give one specific counterexample.)
  - Examples of subsets which are not subspaces are the following:
    - See Exercise 4 in lecture (unit circle)
    - the set of height-3 vectors whose entries are odd numbers
    - the set of height-3 vectors where the sum of entries is equal to 1
From 12a Spanning sets
- Given a subspace W (with a simple description similar to the lecture examples) and a set S, show that S is a spanning set for W.
From 12b Spanning Sets and Linear Independence
- You’ll be given a set S of vectors and asked to determine whether it is linearly independent or linearly dependent. You can use row reduce or use other techniques/theorems (up to you).
- Write down three (or two, four, or five) distinct vectors which are linearly independent. Write down your computation showing that they are linearly independent. (One way to answer this is to use the fact that any subset of the standard basis vectors is linearly independent.)
- Write down three (or two, four, or five) distinct vectors which are linearly dependent. Write down your computation showing that they are linearly dependent. (One quick way to create two linearly dependent vectors is to take a nonzero vector v and a scalar multiple of v like 9v.)
From 13a Bases
- You’ll be given a set S of vectors and asked to determine whether it is a basis for {vectors of height-n}. You can use row reduce technique explained in Lecture 12-13 or use later techniques/theorems (up to you).
- Write down a basis for {vectors of height-4}. (One way to answer this is to write down the appropriate standard basis vectors.)
From 13b Bases (basis computations and dimensions)
- You’ll be given a spanning set of a subspace W and asked to find a basis for W. Use Algorithm 5.
- You’ll be given a matrix M and asked to find a basis for the subspace im(M). Use Algorithm 5, rephrased.
- Write down a matrix M of given size (for example, 4x5) such that the subspace im(M) has a specified dimension (for example, 3). How to answer this: dimension of im(M) = rank(M)
From 14a Basis algorithm for the kernel of a matrix
- You’ll be given a matrix M and asked to find a basis for the subspace ker(M). Use Algorithm 1.
- You’ll be given a matrix M and a number lambda, and asked to find a basis for the lambda-eigenspace of M. Use Algorithm 1.
- Write down a matrix M of given size (for example, 4x5) such that the subspace ker(M) has a specified dimension (for example, 3). How to answer this: ( dimension of ker(M) ) = (number of columns) - ( dimension of im(M) )

Additional problems

Wksheets

Gradescope exercises copied from reading hw. You can see the solution after you click “save answer”.

Note:

You will need to write the quiz on physical paper, so please bring a pen or pencil.
Sheets of scratch paper will be provided.
Know your ID number, since you will need to write it on the exam paper.
The exam papers will be collected, so you don’t need to scan anything.
Calculators are not permitted — no arithmetic simplification is needed.

Exam 1 (Wed, March 9) study guide

Review exercises from lecture 1 through 8 + drawing vectors from lecture 9a, see key, but the exam will only be up to lecture 8b.
Wksheets:
Gradescope exercises. You can see the solution after you click “save answer”.
- Gradescope practice, taken from reading hw 1a, 1b, 2a, 2b, 3a, 3b, 4a, 4b, 5a, 5b.
- Gradescope exercises, taken from reading hw 6a, 6b, 7a, 7b
Lecture videos or lecture notes from lecture 2 to 8b.
Note:
- You will need to write the quiz on physical paper, so please bring a pen or pencil.
- Sheets of scratch paper will be provided.
- Know your ID number, since you will need to write it on the exam paper.
- The exam papers will be collected, so you don’t need to scan anything.
- Calculators are not permitted — no arithmetic simplification is needed.
There will be a total of ten questions:
- 3 or 4 short-answer questions that should take one minute each to answer,
- 3 or 4 questions requiring longer justification or longer computation (row reduce, computing determinant, computing eigenvalues, computing eigenvectors)
  - To compute eigenvalues, you need to know how to compute determinants
  - To compute eigenvectors, you need to know how to perform row reduce on an augmented matrix and use an REF form to describe the vectors using parameters.
- 3 or 4 True/False questions requiring a short justification or a counterexample.