CY-401 Introduction to Linear Algebra - Dec 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
   aProve: Let V be a vector space and W be a subspace of V. Then the map $\eta : V \to V/W$ defined by $\eta(x) = x + W, x \in V$, is a linear transformation.
   How to answer

   • Define quotient space V/W and cosets x+W
   • Verify additivity: eta(x+y)=eta(x)+eta(y)
   • Verify homogeneity: eta(cx)=c eta(x)
   • Conclude eta is linear transformation
2. 7 Marks
   bProve that, if $V_0$ is a subspace of a vector space V, then there exists a subspace $V_1$ of V such that $V = V_0 + V_1$ and $V_0 \cap V_1 = \{0\}$.
   How to answer

   • Recall every subspace has a complement via basis extension
   • Extend basis of V0 to basis of V
   • Define V1 as span of added basis vectors
   • Show V = V0 + V1 and intersection is {0}
3. 7 Marks
   aFind two linear operators T and U on $R^2$ such that $TU = 0$ but $UT \neq 0$.
   How to answer

   • Define T and U by matrices or formulas on basis of R^2
   • Compute composition TU and show it is zero operator
   • Compute UT and show non-zero on some vector
   • Conclude TU=0 but UT != 0
4. 7 Marks
   bFind the characteristic polynomial and the minimal polynomial for the matrix: $$B = \begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$$
   How to answer

   • Compute characteristic polynomial det(B - lambda I)
   • Factor as (2-lambda)^2(3-lambda)
   • Use Jordan block size to determine minimal polynomial
   • Conclude minimal polynomial is (x-2)^2(x-3)
5. 7 Marks
   aIf T be the linear operator on $R^3$ which is represented in the standard basis by the matrix $$A = \begin{bmatrix} 5 & -6 & -6 \\ -1 & 4 & 2 \\ 3 & -6 & -4 \end{bmatrix}$$ Prove that T is diagonalizable. Find the diagonalizable matrix P that $PAP^{-1}$ is diagonal.
   How to answer

   • Compute characteristic polynomial and eigenvalues
   • Find linearly independent eigenvectors (3 in number)
   • Conclude diagonalizable
   • Form P from eigenvectors and verify PAP^{-1} diagonal
6. 7 Marks
   bGiven a $2 \times 2$ matrix: $$B = \begin{bmatrix} 5 & 2 \\ 3 & 4 \end{bmatrix}$$ Use the Cayley-Hamilton theorem to calculate $B^2 - 9B + 14$ and verify that it results in the zero matrix.
   How to answer

   • Compute characteristic polynomial of 2x2 matrix B
   • State Cayley-Hamilton theorem p(B)=0
   • Compute B^2, -9B+14I explicitly
   • Verify sum is zero matrix
7. 7 Marks
   aLet D be a symmetric matrix given by: $$D = \begin{bmatrix} 5 & 2 & -1 \\ 2 & 3 & 2 \\ -1 & 2 & 5 \end{bmatrix}$$ Find the eigenvalues and eigenvectors of D.
   How to answer

   • Compute det(D-lambda I)=0 for symmetric matrix
   • Solve for eigenvalues
   • Solve (D-lambda I)x=0 for eigenvectors
   • Note orthogonality of eigenvectors
8. 7 Marks
   bWhat are the essential properties of inner product spaces? Provide an example of a vector space that satisfies these properties.
   How to answer

   • List inner product axioms: linearity, symmetry, positive-definiteness
   • Explain norm and orthogonality concepts
   • Give example e.g. R^n with dot product or C[a,b]
   • Verify example satisfies axioms
9. 7 Marks
   aConsider the matrix A given by: $$A = \begin{bmatrix} 5 & 4 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{bmatrix}$$

   1. i)Find the Jordan canonical form of the matrix A.
   2. ii)Determine the corresponding Jordan basis.
   How to answer

   • Find eigenvalues 5,1,3 and algebraic multiplicities
   • Check geometric multiplicity / Jordan blocks
   • Write Jordan canonical form J
   • Find Jordan basis vectors solving (A-lambda I) chains
10. 7 Marks
    bShow that the set of all continuous functions on the interval $[0,1]$ equipped with the inner product $$\langle f, g \rangle = \int_0^1 f(x)g(x)dx$$ forms an inner product space. Justify that this definition meets the properties of an inner product.
    How to answer

    • Define vector space of continuous functions on [0,1]
    • Verify linearity and symmetry of integral inner product
    • Verify positive-definiteness using continuity
    • Conclude it is an inner product space
11. 7 Marks
    aLet V be the space of $n \times n$ matrices over a field F, and let A be a fixed $n \times n$ matrix over F. Define a linear operator T on V by $T(B) = AB - BA$. Prove that if A is a nilpotent matrix, then T is a nilpotent operator.
    How to answer

    • Define T(B)=AB-BA and nilpotency A^k=0
    • Expand T^m using binomial-type commutator formula
    • Show T^{2k-1}=0 using A^k=0
    • Conclude T is nilpotent operator
12. 7 Marks
    bLet C be a $5 \times 5$ matrix with characteristic polynomial $(t-1)(t-2)^2(t-3)^2$. Determine the possible Jordan canonical forms of C.
    How to answer

    • Note eigenvalues 1,2,2,3,3 from characteristic polynomial
    • List partitions for eigenvalue 2 and 3 multiplicities
    • Combine block size possibilities
    • Enumerate distinct Jordan forms up to permutation
13. 7 Marks
    aLet $T : C^3 \to C^3$ be a linear transformation with the matrix representation: $$A = \begin{bmatrix} 2 & -i & 0 \\ i & 4 & -3i \\ 0 & 3i & 1 \end{bmatrix}$$ Determine whether T is Hermitian.
    How to answer

    • Recall Hermitian means A* = A (conjugate transpose)
    • Compute conjugate transpose of given A
    • Compare entries, e.g. (1,2) vs (2,1)
    • Conclude whether Hermitian or not
14. 7 Marks
    bSuppose $T: C^2 \to C^2$ is a linear transformation with the matrix representation: $$A = \begin{pmatrix} 1/\sqrt{2} & -i/\sqrt{2} \\ i/\sqrt{2} & 1/\sqrt{2} \end{pmatrix}$$ Determine whether T is unitary or not.
    How to answer

    • Recall unitary means A* A = I
    • Compute conjugate transpose A*
    • Multiply A* A and check identity
    • Conclude unitary or not
15. 14 Marks
    Write short note on any two:

    1. a)Direct sum decompositions
    2. b)Unitary and normal linear transformation
    3. c)Annihilator of a subspace
    How to answer

    • Define chosen topics with key definitions
    • State main theorems/properties
    • Give illustrative example
    • Write concise structured note for each of the two topics