CY-401 Introduction to Linear Algebra - Nov 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
   aShow that $u_1 + u_2 + u_3$ is not a direct sum if $u_1 = \{(x,y,0) \in F^3 \text{ where } x,y \in F\}$ $u_2 = \{(0,0,z) \in F^3 \text{ where } z \in F\}$ $u_3 = \{(0,y,y) \in F^3 \text{ where } y \in F\}$
   How to answer

   • Define direct sum condition: intersection trivial and uniqueness of representation
   • Identify elements of $u_1, u_2, u_3$ as given subsets of $F^3$
   • Show non-trivial intersection e.g. $(0,y,0)$ or dependence among $u_1,u_2,u_3$
   • Conclude sum is not direct
2. 7 Marks
   bLet F be a field of complex numbers and let $T : F^3 \to F^3$ defined by $T(x_1, x_2, x_3) = (x_1 - x_2 + 2x_3, 2x_1 + x_2 - x_3, x_1 - 2x_2)$. Verify that T is linear transformation.
   How to answer

   • Recall definition of linear transformation: additivity and homogeneity
   • Write $T(x_1,x_2,x_3)$ components as given
   • Verify $T(u+v)=T(u)+T(v)$
   • Verify $T(cu)=cT(u)$ and conclude linear
3. 7 Marks
   aSuppose V is finite dimensional and U is subset to V. Show that $U=\{0\}$ if and only if $U^0 = V'$.
   How to answer

   • Define annihilator $U^0$ and dual space $V'$
   • Prove if $U=\{0\}$ then every functional annihilates $U$ so $U^0=V'$
   • Prove converse using finite dimension and dimension formula $\dim U + \dim U^0 = \dim V$
   • Conclude equivalence
4. 7 Marks
   bWrite short note on following:

   1. i)Characteristic values and characteristic vectors
   2. ii)Algebraic multiplicity and Geometric multiplicity
   How to answer

   • Define characteristic value and characteristic vector with $Tx=cx$ / $|A-cI|=0$
   • Give example of computation
   • Define algebraic multiplicity as multiplicity in characteristic polynomial
   • Define geometric multiplicity as dimension of eigenspace and relation
5. 7 Marks
   aLet A be $4 \times 4$ 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 it is also the minimal polynomial.
   How to answer

   • Compute characteristic matrix $A-xI$ and determinant
   • Show determinant equals $x^2(x-1)^2$
   • Test annihilating polynomials to find minimal polynomial
   • Show no lower degree polynomial annihilates A, so minimal equals characteristic
6. 7 Marks
   bLet T be a linear operator on the $n$-dimensional vector space V and suppose that T has $n$-distinct characteristic values. Prove that T is diagonalizable.
   How to answer

   • Assume $n$ distinct characteristic values $c_1,...,c_n$
   • Take corresponding eigenvectors and show linear independence
   • Show $n$ independent eigenvectors form basis of $n$-dimensional $V$
   • Conclude matrix of $T$ in this basis is diagonal
7. 14 Marks
   Find the eigen values and eigen vectors of $A = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}$
   How to answer

   • Form characteristic equation $|A-\lambda I|=0$
   • Solve cubic for eigenvalues
   • Solve $(A-\lambda I)X=0$ for each eigenvalue
   • Normalize/write eigenvectors
8. 7 Marks
   aExplain invariant subspace with suitable examples.
   How to answer

   • Define invariant subspace: $T(W) \subseteq W$
   • Give examples e.g. eigenspace, kernel, range
   • Give matrix example with block form
   • Explain significance
9. 7 Marks
   bFind Jordon canonical form of the matrix $A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 2 & 1 \\ 0 & 1 & 3 \end{bmatrix}$.
   How to answer

   • Compute characteristic polynomial of $3\times 3$ matrix
   • Find eigenvalues and algebraic multiplicities
   • Find eigenvectors and Jordan block sizes
   • Write Jordan canonical form $J$
10. 14 Marks
    If $T \in A(V)$ has all it's characteristic roots in F, then there exists a basis of V such that matrix representation of T is triangular. Prove it.
    How to answer

    • Assume all characteristic roots in $F$
    • Use induction on dimension and existence of eigenvector
    • Extend to basis and consider quotient/restriction
    • Obtain upper triangular matrix representation
11. 7 Marks
    aFind all Skew-symmetric bilinear forms on $R^3$.
    How to answer

    • Define skew-symmetric bilinear form: $f(u,v)=-f(v,u)$
    • Represent by skew-symmetric matrix $A^T=-A$
    • Parametrize $3\times 3$ skew matrices with 3 parameters
    • Write general form $f(x,y)=x^TAy$
12. 7 Marks
    bFind all bilinear forms on the space $n \times 1$ matrices over R which are invariant under $O(n, R)$.
    How to answer

    • Define invariance under $O(n,R)$: $f(Ox,Oy)=f(x,y)$
    • Represent bilinear form by matrix $A$
    • Impose $O^TAO=A$ for all orthogonal $O$
    • Conclude $A=cI$, forms are scalar multiples of dot product
13. 14 Marks
    Write short note on the following (any two):

    1. a)Quotient spaces
    2. b)Direct sum decomposition
    3. c)Self adjoint
    4. d)Jordan blocks
    How to answer

    • Choose any two topics from four options
    • Give definition and key properties for each
    • Give example or theorem
    • Give diagram/matrix illustration where relevant