Midterm Review Notes for Applied Linear Algebra Midterm: Oct 28, 2019 in class
The midterm will focus on testing you on the more theoretical aspects of applied linear algebra. These are concepts that were covered on your quizzes and problem sets, but on the exam you will be asked to show your logic and work in a way that may not have been required for the problem sets. When showing something is true, make sure you state why you can take certain steps. I have provided hints on the practice questions for what assumptions you need in order to show or demonstrate what the question asked for.
You should know the definitions of and how to check:
1. whether a subset of vectors is a subspace
2. linear independence of vectors
3. range, basis, column space, row space, nullspace, and rank
4. properties of a norm
5. conditioning of Ax = b
6. orthogonal & orthonormal vectors
7. Cauchy-Schwarz and triangle inequalities
8. orthogonality of a matrix
You should know the geometric interpretations of (be able to draw in 2-D):
1. linearly independent vectors
2. normalized vectors
3. orthonormal vectors
4. vector projections
5. induced matrix norms for 2-norm
You should know the procedure for:
2. row reduction in terms of elementary matrices (you won’t have to do a full matrix)
3. finding the range, basis, column space, row space, nullspace, and rank of a matrix
4. finding the LU decomposition
5. solving Ax= b given PA =LU
1 6. using elementary matrices to preform Gaussian elimination and form L, perform row exchanges, and multiply a row by a constant.
You may need to use the following properties to demonstrate properties:
1.( AB)T = BT AT
2. det(AB) = det(A)det(B)
3. the determinant of an upper or lower triangular matrix is the product of the diagonal entries.
Here are some practice questions:
1. Find a basis for the row space of the following matrix. What is the rank of A? Find a basis for the column space of A.
1 0 1 0 0 1 0 1 A = (1) 1 1 1 1 0 0 1 1
2. How many solutions would Ax = b have (for problem 1) if
1 1 b1 = ? (2) 1 1
What if 1 0 b2 = ? (3) 1 1
3. By changing one entry of A (problem 1), create a new matrix B that has a unique solution for Bx = b (problem 2, either b).
4. Find a basis for the subspace S ∈ R3 defined by the equation x + 2y + 3z = 0. Verify T that y1 = [−1, 1, 1] ∈ S and find a basis for S that includes y1
5. If v1, v2, and v3 are linearly independent, for what values of c are the vectors v2 − v1, cv3 − v2, and v1 − v3 linearly independent. Hint: To show this, you will need to use the definition of linear independence twice and some algebraic manipulation.
6. The left nullspace of a matrix A is the set of all vectors x such that xT A = 0.
2 1 6 2 (a) Find the left nullspace of A = −1 3 0 Hint: Find the equations through −2 15 2 matrix multiplication for a vector x, then reformulate into a linear system you know how to deal with. (b) Find the nullspace of AT . (c) Show that if A is an n × n matrix, the left nullspace of A is equal to the nullspace of AT . Hint: You will only need the definition of nullspace and the useful property of transposes ((AB)T = BT AT ). To prove this you would need to show both directions (ie left nullspace of A is equal to nullspace of AT and nullspace of AT is equal to left nullspace of A).
(Note: to make the exam feasible I will give you most matrices in reduced row echelon form. However, I may ask that you reduce at least one by hand. So you should decide how much time you want to spend practicing row reduction vs using Matlab)
1 7. Verify the Cauchy-Schwarz and triangle inequalities for the vectors x = −1, 2 2 y = 3 −4
8. Draw a sketch of two linearly independent vectors in R2 that are normalized, but not orthogonal. Be precise in giving the vectors length and direction.
9. Draw a sketch illustrating a vector projection of one vector onto another. Label with formula the distance projected, the direction of the projection, and the projected vector itself. Your formula should only be a function of the original vectors (not the angle between them).
10. Show that if x and y are vectors in Rn, then hAx, yi = hx, AT yi. (Hint: you should only need the definition of inner product, the property of the transpose of a matrix product, and rules of matrix multiplication.)
11. Show that if A and B are orthogonal matrices, then AB and BA are orthogonal matrices. (Hint: You should only need the definition of orthogonal matrices and the rule for transpose of a matrix product.)
12. (6.25) Show that if P is an orthogonal matrix, and x and y are vectors in Rn, then hPx, Pyi = hx, yi. (Hint: You should only need the definition of orthogonal matrices, the vector/matrix multiplication formulation of inner product, and the rule for transposes of products of matrices.)
13. Show that ||x||1 is a vector norm by verifying Def. 7.1 in the text. (Hint: you should only need the Def. 7.1, algebra with summations, and an upper bound on the absolute value as used in the proof in Theorem 7.1 Triangle Inequality.)
3 14. Show that for I an n × n identity matrix:
(a) ||I||2 = 1 p (b) ||I||F = (n) 15. Draw a conceptual sketch of the matrix induced 2-norm, as in Fig. 7.8. Label ||x||2 = 1, ||Ax||2, and ||A||2 for some A. What is the semi-major axis? 16. If you have PA = LU how can you calculate the determinant of A
17. List the steps to find x given PA = LU for Ax = b. 2 1 0 18. What two elementary row matrices E21(t1) and E32(t2) put A = 6 4 2 into 0 3 5 upper triangular form? E21(t1)E32(t2)A = U. Calculate L using these matrices (use the properties of inverse and multiplication of elementary matrices).
19. Construct a 4 × 4 matrix that will take 3rd column of another 4 × 4 matrix, A, multiply it by 4, and subtract it from the 2nd column of A, through matrix multiplication.
20. Take the inverse of the elementary matrix you constructed in the previous problem.
21. Construct a 3x3 matrix, P1 that swaps the 2nd and 3rd row of another 3x3 matrix.
22. What is the inverse of P1? Show whether P1 is an orthogonal matrix.
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