Homework 1

Homework 1

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 typewritter 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 nessesary source files if there are any) in addition to the PDF file.

Questions

1. A vector space defined on the field of the real numbers R is a set V, equipped with a vector addition operator (vector + vector) and a scalar multiplication (scalar times vector) with the following properties

(1) Commutativity: u+v=v+uforu,v∈V.

(2) Associativity: u+ (v+w) = (v+u) +w for allu,v,w∈V. (3) Identity element: There exists an elementz∈V such that for every

u∈V we have u+z=u. ( zis called the identity element and is often denoted by0).

(4) Inverse: for every u ∈ V there existu⁰ ∈V such that u+u⁰ = z (where the identity element zwas defined above. The vector u⁰ is called the (additive) inverse of uand is usually denoted by u⁻¹ or

−u.)

1

Linear Algebra for Computer Science and Engineering Fall 2022

Behrooz Nasihatkon

(5) 1u=u. (notice that 1∈R.)

(6) a(bu) = (ab)ufor alla, b∈Randu∈V. (7) a(u+v) =au+avfor alla∈Randu,v∈V. (8) (a+b)u=au+bufor alla, b∈Randu∈V.

Prove the following. In each step state which of the above axioms (1-8) you are using. You might also use the previous results, i.e. to prove d you may use parts a,b,c. Notice that there is no notion of subtraction (−) here.

(a) Ifzis an identity element, thenz+u=ufor allu∈V. (notice that (3) statesu+z=u.)

(b) The identity elementzis unique.

(c) The inverse of the identity element is itself.

(d) The inverse of any vector is unique.

(e) 0u=zfor the scalar 0∈R. (f) az=zfor any scalara∈R.

(g) (−1)u=u⁰ where u⁰ is the inverse element ofu.

2. Consider a matrixA∈R^m×n, such thatAx= 0 for allx∈R. Prove that A=0_m×n, that is all the entries ofAare zero.

3. Consider a matrix A∈R^m×n, such thatAx_i = 0 forx₁,x₂, ...,x_n ∈Rⁿ, wherex₁,x₂, . . . ,x_n form a basis forRⁿ. Prove thatA= 0_m×n.

4. Consider a square matrix A ∈ R^n×n for which Ax = x for all x ∈ Rⁿ. Prove thatA=In, thenbynidentity matrix.

5. Give an example of a matrixA∈R, such thatAx=x for some nonzero vectorx∈Rⁿ,Ais not the identity matrix.

6. Assume that the vectors a₁,a₂, ...,a_n are linearly independent. Prove that the set of vectors a⁰₁,a₂, ...,a_n are also linearly independent where a⁰₁=a₁+βa₂ for some scalarβ.

7. The dot product can also be defined on matrices. Consider two matrices A,B∈R^m×n. Their dot product can be defined ashA,Bi=Pm

i=1

Pn

j=1AijBij. (a) Prove that hA,Bi = trace(A^TB) = trace(B^TA) = trace(AB^T), where

trace(S) =P

iS_ii gives the sum of the diagonal elements of a square matrixS.

(b) Prove thathAB,Ci= B,A^TC

= A,CB^T

Hint: (AB)^T =B^TA^T. 8. Prove that a triangular matrix with at least one zero diagonal element is

singular.