CY-401 Introduction to Linear Algebra - Jun 2024 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
   aLet $U_1, U_2$ and $U_3$ be subspaces of a vector space $V$. Prove that $$U_1 \cap (U_2 + (U_1 \cap U_3)) = (U_1 \cap U_2) + (U_1 \cap U_3)$$
   How to answer

   • Define sum and intersection of subspaces
   • Show LHS subset RHS using decomposition of element
   • Show RHS subset LHS
   • Conclude modular law equality
2. 7 Marks
   bLet $V$ be a vector space. Show that $\dim(V) = n \ge 1$ if and only if there exist one-dimensional subspaces $U_1, \dots, U_n$ such that $V = U_1 \oplus \dots \oplus U_n$.
   How to answer

   • Recall dim(V)=n means basis of n vectors
   • Construct one-dimensional spans of basis vectors
   • Show sum is direct using linear independence
   • Prove converse using dimension of direct sum
3. 7 Marks
   aLet $V$ be a vector space over a field $F$ and $T : V \to V$ be a linear transformation. Define the annihilator of a subspace $W$ of $V$ as $$W^0 = \{ f \in V^* \mid f(w) = 0 \text{ for all } w \in W \},$$ Where $V^*$ is the dual space of $V$. Prove that $W^0$ is a subspace of $V^*$ and determine its dimension in terms of the dimension of $V$ and $W$.
   How to answer

   • Define dual space $V^*$ and annihilator $W^0$
   • Verify zero functional in $W^0$ and closure under addition and scalar multiplication
   • Extend basis of $W$ to $V$ and construct dual basis
   • Conclude $\dim W^0 = \dim V - \dim W$
4. 7 Marks
   bLet $A$ be the $4 \times 4$ real matrix $$A = \begin{bmatrix} 1 & 1 & 0 & 0 \\ -1 & -1 & 0 & 0 \\ -2 & -2 & 2 & 1 \\ 1 & 1 & -1 & 0 \end{bmatrix}$$ Show that the characteristic polynomial for $A$ is $x^2(x-1)^2$ and that is also the minimal polynomial.
   How to answer

   • Compute $\det(A - xI)$ by expansion
   • Simplify to $x^2(x-1)^2$
   • Check annihilating polynomials on $A$
   • Show no lower-degree polynomial annihilates $A$, hence minimal polynomial
5. 7 Marks
   aLet $W$ be an invariant subspace for linear operator $T$. Prove that the minimal polynomial for the restriction operator $T_W$ divides the minimal polynomial for $T$, without referring to matrices.
   How to answer

   • Define invariant subspace and restriction operator $T_W$
   • Take minimal polynomial $m_T$ of $T$ so $m_T(T)=0$
   • Restrict to $W$ to get $m_T(T_W)=0$
   • Conclude minimal polynomial of $T_W$ divides $m_T$
6. 7 Marks
   bFor the matrix $A$: $$A = \begin{bmatrix} 3 & 1 \\ 2 & 2 \end{bmatrix}$$ Verify Cayley-Hamilton theorem by finding its characteristic polynomial and substituting the matrix into it.
   How to answer

   • Compute characteristic polynomial $\det(A-\lambda I)$
   • Find $p(A)$ by substituting matrix $A$
   • Verify matrix multiplication gives zero matrix
   • Conclude Cayley-Hamilton verified
7. 7 Marks
   aConsider the matrix $A$: $$A = \begin{bmatrix} 6 & 1 & 0 \\ 0 & 6 & 1 \\ 0 & 0 & 6 \end{bmatrix}$$ Find the Jordan canonical form of the matrix $A$ and the corresponding Jordan basis.
   How to answer

   • Find eigenvalue $6$ with algebraic multiplicity $3$
   • Compute $\dim \ker(A-6I)$ to get number of Jordan blocks
   • Conclude single $3\times 3$ Jordan block is the form
   • Solve $(A-6I)$ chain for generalized eigenvectors for Jordan basis
8. 7 Marks
   bLet $V$ be an inner product space, and let $\alpha$ and $\beta$ be vectors in $V$. Show that $\alpha = \beta$ if and only if $(\alpha \mid \gamma) = (\beta \mid \gamma)$ for every $\gamma$ in $V$.
   How to answer

   • Assume $\alpha = \beta$ then inner products equal
   • Conversely assume $(\alpha|\gamma)=(\beta|\gamma)$ for all $\gamma$
   • Take $\gamma = \alpha-\beta$ and use linearity
   • Use positive-definiteness to get $\alpha-\beta=0$
9. 7 Marks
   aConsider a matrix $A$ given by: $$A = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix}$$

   1. i)Find the eigen values and corresponding eigen vectors of A.
   2. ii)Determine the invariant subspaces corresponding to each distinct eigen value.
   How to answer

   • Compute characteristic equation $\det(A-\lambda I)=0$
   • Solve for eigenvalues $4$ and $2$ with eigenvectors
   • Describe eigenspaces as spans
   • List invariant subspaces for each eigenvalue
10. 7 Marks
    bGiven that the set $\{(1, 0, 0) (0, 2, 1) (2, 0, 1)\}$ is a basis of $\mathbb{R}^3$. If $T : \mathbb{R}^3 \to \mathbb{R}^3$ is a linear transformation such that $T(1, 0, 0) = (0, 0, 1)$, $T(0, 2, 1) = (1, 2, 0)$ and $T(2, 0, 1) = (1, 1, 1)$. Find $T(2, -3, 4)$.
    How to answer

    • Express $(2,-3,4)$ as linear combination of given basis
    • Solve for coordinates
    • Apply linearity $T(\sum c_i v_i)=\sum c_i T(v_i)$
    • Compute $T(2,-3,4)$
11. 7 Marks
    aIf $V$ is the space of all polynomials of degree less than or equal to $n$ over a field $F$, prove that the differentiation operator on $V$ is nilpotent.
    How to answer

    • Define $V=P_n(F)$ of dimension $n+1$
    • Show differentiation $D$ is linear
    • Compute $D^{n+1}=0$ on basis $1,x,\dots,x^n$
    • Conclude $D$ is nilpotent of index $n+1$
12. 7 Marks
    bConsider a $2 \times 2$ matrix $G$ with characteristic polynomial $(t + 1)^2(t - 2)$ and minimal polynomial $(t + 1)(t - 2)$. Determine the primary decomposition of $R^2$ with respect to $G$.
    How to answer

    • Factor characteristic and minimal polynomials
    • Identify primary components for $(t+1)$ and $(t-2)$
    • Use minimal polynomial to get dimensions of blocks
    • Write $R^2$ as direct sum of $G$-invariant primary subspaces
13. 7 Marks
    aLet $A, B \in O(n)$, the orthogonal group of real $n \times n$ matrices. Define the bilinear form as: $$B(A, B) = tr(A^T B)$$ Verify if $B$ preserves the group structure of $O(n)$.
    How to answer

    • Recall $O(n)=\{A|A^TA=I\}$
    • Define $B(A,B)=tr(A^TB)$ and check bilinearity
    • Test closure and inner product properties on $O(n)$
    • Discuss whether group structure is preserved
14. 7 Marks
    bWrite a detail note on skew symmetric bilinear forms.
    How to answer

    • Define skew-symmetric condition $B(u,v)=-B(v,u)$
    • Give matrix condition $A^T=-A$
    • State properties and canonical form
    • Give examples
15. 14 Marks
    Write short note on any two:

    1. a)Cayley-Hamilton theorem
    2. b)Primary decomposition theorem
    3. c)Invariant subspaces
    How to answer

    • State Cayley-Hamilton theorem with characteristic polynomial
    • State primary decomposition theorem with invariant direct sum
    • Define invariant subspaces with examples
    • Conclude with significance