CY-401 Introduction to Linear Algebra - Jun 2023 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aSuppose V is finite dimensional and U is subspace of V. Show that $U = V$ if and only if $U^0 = \{0\}$.
   How to answer

   • Define annihilator U^0 in dual space V*.
   • If U=V then only zero functional vanishes on all V.
   • Conversely if U is proper, extend basis of U to V and construct non-zero functional vanishing on U.
   • Conclude U=V iff U^0={0} using finite-dimensionality.
2. 7 Marks
   bShow that the mapping $T : V_3(R) \to V_3(R)$ defined as $T(a_1, a_2, a_3) = (3a_1 - 2a_2 + a_3, a_1 - 3a_2 - 2a_3)$ is a linear transformation from $V_3(R)$ into $V_2(R)$.
   How to answer

   • Check T maps V_3(R) to V_2(R).
   • Verify additivity T(alpha+beta)=T(alpha)+T(beta).
   • Verify homogeneity T(c alpha)=cT(alpha).
   • Conclude T is linear transformation.
3. 7 Marks
   aSuppose V is finite dimensional and U is a subspace of V. Then prove that $\dim\left(\frac{V}{M}\right) = \dim V - \dim U$.
   How to answer

   • Define quotient space V/U and cosets.
   • Extend basis of U to basis of V.
   • Show cosets of added vectors form basis of quotient.
   • Count dimensions to get dim(V/U)=dimV-dimU.
4. 7 Marks
   bWrite short note on :

   1. i)Cayley-Hamilton theorem
   2. ii)Annihilating Polynomials
   How to answer

   • State Cayley-Hamilton theorem: every matrix satisfies its characteristic polynomial.
   • Give implication / brief proof idea.
   • Define annihilating polynomial p(T)=0.
   • Give examples and relation to minimal polynomial.
5. 7 Marks
   aLet $a, b, c$ be elements of a field F, and let A be the following $3 \times 3$ matrix over F, $A = \begin{bmatrix} 0 & 0 & c \\ 1 & 0 & b \\ 0 & 1 & a \end{bmatrix}$ prove that the characteristic polynomial for A is $x^3 - ax^2 - bx - c$ and that it is also the minimal polynomial for A.
   How to answer

   • Compute det(xI-A) for given companion matrix.
   • Expand to get $x^3-ax^2-bx-c$.
   • Show no lower degree polynomial annihilates A by direct computation.
   • Conclude characteristic equals minimal polynomial.
6. 7 Marks
   bLet A be a $n \times n$ triangular matrix over the field F. Prove that characteristic values of A are the diagonal entries of A. (i.e., the scalars $A_{ii}$).
   How to answer

   • Write det(A - lambda I) for triangular matrix.
   • Observe determinant is product of diagonal terms (A_ii - lambda).
   • Identify roots as diagonal entries.
   • Conclude characteristic values are A_ii.
7. 14 Marks
   Diagonalize the following matrices $\begin{bmatrix} 1 & 0 & -1 \\ 1 & 2 & 1 \\ 2 & 2 & 3 \end{bmatrix}$.
   How to answer

   • Compute characteristic polynomial det(A-lambda I).
   • Find eigenvalues and eigenvectors.
   • Form invertible P from eigenvectors.
   • Write P^{-1}AP as diagonal matrix.
8. 7 Marks
   aShow that $V_2(R)$ is an inner product space defined by $(\alpha, \beta) = 3a_1b_1 + 2a_2b_2 \forall, \alpha = (a_1, a_2), \beta = (b_1, b_2) \in V_2(R)$
   How to answer

   • Verify linearity and symmetry of given form.
   • Check positivity: (alpha,alpha)>=0.
   • Check definiteness: zero only for zero vector.
   • Conclude inner product space axioms hold.
9. 7 Marks
   bExplain Jordan blocks with suitable examples.
   How to answer

   • Define Jordan block J_k(lambda) structure.
   • Explain ones on superdiagonal.
   • Give 2x2 and 3x3 examples.
   • Mention role in Jordan canonical form.
10. 14 Marks
    Show that "Two nilpotent linear transformations are similar if and only if they have same invariants".
    How to answer

    • Define nilpotent transformation and its invariants / partitions.
    • Prove similar transforms have same invariants.
    • Prove same invariants imply same Jordan form.
    • Conclude similarity iff same invariants.
11. 7 Marks
    aFind a basis for the space of all skew-symmetric linear forms on $R^n$.
    How to answer

    • Define skew-symmetric condition f(alpha,beta)=-f(beta,alpha).
    • Construct basis forms f_ij with matrix E_ij-E_ji.
    • Show spanning and linear independence.
    • Count dimension n(n-1)/2.
12. 7 Marks
    bFind all bilinear forms on the space of $n \times 1$ matrices over C which are invariant under $O(n, c)$.
    How to answer

    • Define invariance under O(n,c): f(gx,gy)=f(x,y).
    • Use orthogonal group action on matrices.
    • Show only scalar multiples of standard dot product survive.
    • List all invariant bilinear forms.
13. 14 Marks
    Write short note on following (any two):

    1. a)Dual Spaces
    2. b)Primary decomposition theorem
    3. c)Bilinear forms
    4. d)Inner product spaces
    How to answer

    • Define chosen topics with key theorems.
    • Give properties and examples.
    • Include relevant formulas / structures.
    • Summarize applications in linear algebra.