Homework 2

Homework 2

Read these first:

i You may write your solutions on paper, under a word processing software (MS-word, Libre Office, etc.), or under L^ATEX.

ii If writing on paper, you must use a scanner device or a Camera Scanner (CamScanner) software to scan the document and submit a single PDF file.

iii A 10% extra score will be given to solutions written under L^ATEX, provided that you follow either of the following conventions:

(a) Represent scalars with normal (italic) letters (a, A), vectors with bold lower-case letters (a, using \mathbf{a}), and matrices with bold upper-case letters (A, using\mathbf{A}).

(b) Represent scalars with normal (italic) letters (a, A), vectors with bold letters (a, A), and matrices with typewriter upper-case letters (A, using\mathtt{A}).

(c) In all cases, you must submit a singlePDF file.

(d) If writing under L^ATEX, you must submit the.tex source (and other necesary source files if there are any) in addition to the PDF file.

Questions

Basis

1. LetS ⊂Rⁿ be astrict linear subspace ofRⁿ (strict meaningS 6=Rⁿ or dim(S)< n). I argue that the set of standard basis vectorse₁,e₂, . . . ,e_n∈ Rⁿ form a basis forS because

• e₁,e₂, . . . ,e_n are linearly independent, and

• evey vector isScan be written as a linear combination ofe1,e2, . . . ,en. How is my argument wrong?

1

Linear Algebra for Computer Science and Engineering Fall 2022

Behrooz Nasihatkon

Linear Equations

To answer the following questions you need to use the fact that the set of solutions to a system of linear equations Ax= bis in the form of {x_p+x_n | x_n∈ N(A)}, wherex_p is a particular solution.

2. LetA∈R^m×nbe a fat matrix (i.e. m < n) withfull row rankandb∈R^m. Show thatAx=bhas infinitely many solutions.

3. Let A∈ R^m×n be a tall matrix (i.e. m > n) with full column rank and b∈R^m. Show thatAx=bhas either no solution or exactly one solution.

4. LetA∈R^m×nbe rank-deficient (rank(A)<min(m, n)) andb∈R^m. Show Ax=bhas either no solution or infinitely many solutions.

5. Consider the system of linear equations Ax = b with A ∈ R^m×n and b∈R^m, and letS be the set of solutions to it. Show that

(a) S is a linear subspace if and only ifb=0.

(b) IfS is nonempty, then there exists a vectory∈Rⁿ such that the set {z−y|z∈ S}is a linear subspace.

Projections

6. Consider a linear subspace S and a vector y ∈ S. Using the projection formula, show that the projection ofyinto S is itself.

7. For a linear subspace S ⊆ Rⁿ its orthogonal complement is defined as S^⊥ ={y ∈Rⁿ | y^Tx= 0 for allx ∈ S}. In other words, S^⊥ comprises all the vectors that are perpendicular to all vectors in S. Show that the orthogonal complement of a linear subspace is a linear subspace.

8. Prove that thenull spaceof a matrix is the orthogonal complement of its row space.

9. Let P ∈R^n×n be the projection matrix into a linear subspace S. Show thatI−Prepresents the projection into the orthogonal complement ofS.

Hint: First show thatI−Pis a projection matrix. Then, show that any vectory∈ S^⊥ can be written as (I−P)xfor some x∈Rⁿ.

10. Show that rank(I−P) =n−rank(P) for a projection matrixP∈R^n×n.

Determinant

11. Prove that the determinant of an orthogonal matrix is either equal to 1 or−1.

12. Show that the determinant of a projection matrix is either equal to 0 or 1. How do you explain this geometrically?