CY-401 Introduction to Linear Algebra - Jun 2025 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
   aDefine subspace of a vector space. Show that the quotient space of finitely generated space is finitely generated.
   How to answer

   • Define subspace: non-empty subset closed under addition and scalar multiplication
   • Define quotient space V/W and cosets
   • Take finite spanning set of V, consider their cosets in V/W
   • Show cosets span V/W, hence finitely generated
2. 7 Marks
   bState and prove Rank nullity theorem.
   How to answer

   • State rank-nullity: rank(T)+nullity(T)=dim V
   • Consider basis of kernel and extend to basis of domain
   • Apply T to extended basis vectors to get basis of range
   • Count dimensions to prove formula
3. 7 Marks
   aLet $f$ be linear functional on $R^3$. Show that $\exists \bar{a} \in R^3$ such that $f(\bar{r}) = \bar{r} \cdot \bar{a}$.
   How to answer

   • Define linear functional on $R^3$
   • Write $f(e_i)=a_i$ for standard basis
   • Express $\bar{r}=x_1e_1+x_2e_2+x_3e_3$ and use linearity
   • Identify $\bar{a}=(a_1,a_2,a_3)$ so $f(\bar{r})=\bar{r}\cdot\bar{a}$
4. 7 Marks
   bDefine Characteristic polynomials. Show that all eigen values of $A^*A$ are real and it is unitary similar to diagonal matrix.
   How to answer

   • Define characteristic polynomial $det(A-\lambda I)$
   • Show $A^*A$ is Hermitian / positive semi-definite
   • Deduce eigenvalues $\lambda = \|Ax\|^2/\|x\|^2 \ge 0$ real
   • Use spectral theorem to get unitary diagonalization
5. 7 Marks
   aState and prove Cayley Hamilton theorem.
   How to answer

   • State Cayley-Hamilton: every matrix satisfies its characteristic equation
   • Consider $adj(xI-A)(xI-A)=p(x)I$ argument
   • Expand and equate coefficients of powers of A
   • Conclude $p(A)=0$
6. 7 Marks
   bLet T be a linear transformation on a vector space V of dimension $n$. Suppose that T has distinct eigen values. Then show that T is diagonalizable.
   How to answer

   • Assume $n$ distinct eigenvalues $\lambda_1,...,\lambda_n$
   • Take corresponding eigenvectors $v_i$
   • Prove eigenvectors for distinct eigenvalues are linearly independent
   • n independent vectors form basis, hence T diagonalizable
7. 7 Marks
   aDefine adjoint of Linear Transformation. The linear transformation $T:R^3 \to R^4$ defined by $T(x,y,z) = (x-2y+z, 2x+y+2z, 3x-y+z)$ then show that T is a linear transformation.
   How to answer

   • Define adjoint $T^*$ via $<T(x),y>=<x,T^*(y)>$
   • Verify additivity $T(u+v)=T(u)+T(v)$ using components
   • Verify homogeneity $T(cu)=cT(u)$
   • Conclude T is linear transformation
8. 7 Marks
   bLet W be a subspace of finite dimensional inner product space V and $x \in V$ such that $<x,y> + <y,x> \le <y,y>$ for all $y \in W$. then show that $x \in W^{\perp}$?.
   How to answer

   • Recall $W^{\perp}=\{x: <x,y>=0 \forall y\in W\}$
   • Use given inequality $<x,y>+<y,x> \le <y,y>$ for all $y\in W$
   • Replace $y$ by $ty$ / scalar multiples and let $t\to 0$
   • Deduce $<x,y>=0$, so $x\in W^{\perp}$
9. 7 Marks
   aShow that there is no proper open subspace of an inner product space.
   How to answer

   • Assume W is open subspace containing non-zero vector
   • Use openness: ball around vector stays in W
   • Use subspace scaling property for arbitrary scalars
   • Show W must be whole space, no proper open subspace
10. 7 Marks
    bDefine Canonical form of linear transformation. Show that every $m \times n$ matrix is equivalent to unique matrix in one of canonical form.
    How to answer

    • Define canonical / normal form under equivalence
    • Define equivalence $B=PAQ$ with invertible P,Q
    • Show rank is invariant and determines form
    • Construct $diag(I_r,0)$ unique canonical form
11. 7 Marks
    aShow that all eigen values of hermitian matrix are all real.
    How to answer

    • Define Hermitian $A^*=A$
    • Take $Ax=\lambda x$ with $x\ne 0$
    • Consider $<Ax,x>=\lambda\|x\|^2$ and its conjugate
    • Equate to get $\lambda=\bar{\lambda}$, hence real
12. 7 Marks
    bDefine Jordan canonical form. Reduce the matrix $$\begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 3 \end{bmatrix}$$ Into Jordan canonical forms.
    How to answer

    • Define Jordan canonical form and Jordan blocks
    • Find eigenvalues 1,1,3 and characteristic polynomial
    • Determine geometric multiplicity / minimal polynomial for eigenvalue 1
    • Write Jordan form with one $2\times2$ block for 1 and $[3]$
13. 7 Marks
    aShow that a bilinear form on V is a product of linear functional iff it is of rank 1.
    How to answer

    • Define bilinear form and rank 1
    • If $f(x,y)=f_1(x)f_2(y)$ show image is one-dimensional
    • Conversely if rank 1, pick bases to factor form
    • Conclude equivalence
14. 7 Marks
    bState and prove Gram Schmidt orthogonalization process.
    How to answer

    • State Gram-Schmidt: orthonormal set from independent set
    • Construct $v_1'=u_1$, $v_k'=u_k-\sum proj$ inductively
    • Prove orthogonality by induction using inner product
    • Normalize to get orthonormal basis
15. 14 Marks
    Write short notes on

    1. i)Symmetric bilinear forms
    2. ii)Group preserving bilinear forms
    3. iii)Inner product space
    How to answer

    • Define symmetric bilinear form with examples
    • Explain group preserving bilinear form / congruence
    • Define inner product space with axioms
    • Give matrix representation and properties for each