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Summer 2024 Linear Algebra Take-home Final Exam
发布时间：2024-06-20
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Summer 2024 Linear Algebra Take-home Final Exam
Due on Tuesday June 18, 2024 at 11:59 PM
Please show all your work to me.
1. Determine whether the statements below are true or false. (Justify your answers: If a statement is true, explain why it is true; if it is false, explain why, or give a
counterexample for which it is false.)
(a) Two linear systems are equivalent if they have the same solution set.
(b) A linear combination of vectors a1, . . . , an can always be written in the form Ax for a suitable matrix A and a vector x.
(c) The echelon form of a matrix is unique.
2. Let 
(a) Find eigenvalues of A.
(b) Find bases for the eigenspaces of A.
(c) Find an orthogonal matrix U and a diagonal matrix D such that A = UDUt. Briefly explain yourself.
(d) Find ‖A‖F
3. Consider the matrix 
(a) Find a basis ℬ for N(A).
(b) Let V=N(A). Then we can define a linear transformation T: V → R3 by T(v) = Av. Write down the matrices for T in terms of the basis ℬ of Vand the standard basis ℇ = {e1 e2 e3 }.
4. Let 
V, where V = span{v1, v2, v3 }; (b) find A2024 v
5. (a) If A is a nonzero 5 × 3 matrix, what is the maximum possible value of rank(A)?
(b). Give an example of 4 linearly independent vectors in P3, or state that it is not possible to do so.
(c). Give an example of 4 vectors that span P3, or state that it is not possible to do so.
6. Let 
(a). Find all eigenvalues of A.
(b). Find all eigenvalues of B.
(c). Which matrix A orB is diagonalizable?
(d). Diagonalize the matrix stated in (c), i.e. find an invertible matrix P and a diagonal matrix D such that A = PDP −1 orB = PDP −1 .
7. Suppose that A is ann × n matrix such that all eigenvalues of A are positive. Explain why the matrix A + 2024In must be invertible.
8. Suppose that {u1, u2, u3 } is an orthogonal set of vectors in ℝ4 , and ||u1 || = 2, ||, u2 || = 3, and
||u3 || = 10. Let y = 2u1 − 5 u2 + u3. (a). Find ||y||. (b). Find y ∗ u2 .
9. Let A be ann × n orthogonal matrix, and let x andy be vectors in Rn.
(a) Show that (Ax) • (Ay) = x • y.
(b) Use (a) to show that ||Ax|| = ||x||.
10. Let W = Span{x1, x2}, where 
and 
(a) Use the Gram-Schmidt process to find an orthogonal basis{u1, u2 } for W.
Let 
Decompose yas y = 
+ z, where 
is in W and z is in W⊥ .
11. Let
and 
Show that B is invertible and find [B−1AB]2024 .
13. Let 
Find the SVD of A.
14. Which of the matrices that follow are Hermitian? Normal? Explain why.
Bonus 1. Let A ∈ Rm×n with m > n. Show that ifrank(A) = n, then At A is positive definite and AAt is semipositive definite.
Bonus 2. Let A be ann × n Hermitian matrix and let x be a vector in cn. Show that if C = xtAx, thencis real.
Bonus 3. Let U be a unitary matrix. Prove that (a) U is normal. (b) ⅡuxⅡ=ⅡxⅡ for all x ∈cn . (c) if λ is an eigenvalue of U, then |λ| = 1.
Bonus 4. Prove that if A is asymmetric matrix with eigenvalues λ1, λ2 , ..., λn , then the singular values of A are |λ1 |, |λ2 |, ... , |λn |.
Bonus 5. Show that the diagonal entries of a Hermitian matrix must be real.
