CY-401 Introduction to Linear Algebra - Jun 2022 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

   1. i)Direct sum of a vector space
   2. ii)Dual Space of a vector space
   3. iii)Quotient Space of vector space
   4. iv)Annihilator of a sub space
   How to answer

   • Define direct sum: V = W1+W2 with W1 cap W2 = {0}, unique decomposition.
   • Define dual space V* = set of linear functionals on V.
   • Define quotient space V/W with cosets v+W and induced operations.
   • Define annihilator W^0 = {f in V* : f(w)=0 for all w in W}.
2. 7 Marks
   b$W_1$ and $W_2$ are the sub spaces of $R^3$ if $W_1 = \{(0, 0, z) | z \in R\}$ and $W_2 = \{(x, y, x) | x, y \in R\}$ show that $R^3 = W_1 \oplus W_2$
   How to answer

   • Show W1 and W2 are subspaces of R^3.
   • Show W1 cap W2 = {0}.
   • Show dim W1 + dim W2 = 3 or every vector is sum.
   • Conclude R^3 = W1 \oplus W2.
3. 7 Marks
   aShow that the matrix $A = \begin{bmatrix} 3 & -1 & 1 \\ -1 & 3 & -1 \\ 1 & -1 & 3 \end{bmatrix}$ is diagonalizable. Hence find diagonal and transforming matrix.
   How to answer

   • Compute characteristic polynomial |A-λI|=0 and eigenvalues.
   • Find three linearly independent eigenvectors.
   • Form P from eigenvectors and D = P^{-1}AP.
   • Conclude diagonalizable with D and P.
4. 7 Marks
   bFind the characteristic equation of the matrix $A = \begin{bmatrix} 4 & 3 & 1 \\ 2 & 1 & -2 \\ 1 & 2 & 1 \end{bmatrix}$. Hence find $A^{-1}$
   How to answer

   • Compute characteristic determinant |A-λI|.
   • Simplify to characteristic equation cubic.
   • Apply Cayley-Hamilton to express A^{-1} as polynomial in A.
   • Compute adjugate / polynomial to get A^{-1}.
5. 7 Marks
   aFind all the Eigen values and a basis of each Eigen space of the linear operator $T : R^2 \to R^2$ defined by $T(x, y) = (x+2y, 3x+2y)$
   How to answer

   • Write matrix of T relative to standard basis.
   • Solve |A-λI|=0 for eigenvalues.
   • Solve (A-λI)v=0 for each λ to get eigenvectors.
   • State basis of each eigenspace.
6. 7 Marks
   bIf $\alpha$ and $\beta$ are vectors in an inner product space, then show that $\|\alpha+\beta\|^2 + \|\alpha-\beta\|^2 = 2\|\alpha\|^2 + 2\|\beta\|^2$
   How to answer

   • Recall norm from inner product $\|x\|^2 = \langle x,x\rangle$.
   • Expand $\|\alpha+\beta\|^2$ and $\|\alpha-\beta\|^2$ using linearity.
   • Add the two expansions; cross terms cancel.
   • Obtain $2\|\alpha\|^2+2\|\beta\|^2$.
7. 7 Marks
   aState and prove primary decomposition theorem.
   How to answer

   • State primary decomposition: V = direct sum of Ker p_i(T)^{r_i}.
   • Show invariance and coprime polynomial lemma.
   • Prove existence of decomposition by induction.
   • Prove uniqueness / directness.
8. 7 Marks
   b$T : IR^2 \to IR^2$ by $T\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$. Find all T-invariant subspace of $IR^2(R)$.
   How to answer

   • Write matrix A = [[2,-5],[1,-2]] and find characteristic polynomial.
   • Check eigenvalues over R (none / complex).
   • Use that invariant subspaces correspond to eigenvectors.
   • List {0}, R^2 and any 1-dim invariant subspaces if exist.
9. 7 Marks
   aCheck whether the linear operator T defined on $C^2$ by $T(x, y) = (2x + 2y, x + 2y)$ is

   1. i)Self adjoint
   2. ii)Unitary
   How to answer

   • Write matrix of T and its conjugate transpose.
   • Check T* = T for self-adjoint.
   • Check T*T = I for unitary.
   • Conclude which property holds.
10. 7 Marks
    bLet A be an $n \times n$ matrix over K. Show that the mapping defined by $f(x, y) = x^T A y$ is a bilinear form
    How to answer

    • Recall definition of bilinear form: linear in each argument.
    • Expand f(ax1+x2, y) using matrix multiplication.
    • Expand f(x, by1+y2) similarly.
    • Conclude linearity in both variables.
11. 7 Marks
    aFind the Jordan canonical form of the matrix $\begin{pmatrix} 0 & 4 & 2 \\ -3 & 8 & 3 \\ 4 & -8 & -2 \end{pmatrix}$
    How to answer

    • Compute characteristic polynomial and eigenvalues.
    • Find algebraic and geometric multiplicities.
    • Determine sizes of Jordan blocks.
    • Write Jordan canonical form J.
12. 7 Marks
    bProve that if x is a eigen vector of T corresponding to eigen value $\lambda$ then $\alpha x$ is also an eigen vector of T corresponding to same eigen value $\lambda$; $\alpha$ is a non zero scalar.
    How to answer

    • Assume Tx = λx with x ≠ 0.
    • Compute T(αx) = αT(x) by linearity.
    • Substitute to get T(αx) = λ(αx).
    • Note αx ≠ 0 since α ≠ 0, hence eigenvector.
13. 7 Marks
    aLet W be the subspace of $IR^3(C)$ spanned by $(0, 1, -1)$ and $(1, -1, -1)$. Find a basis of A(W).
    How to answer

    • Take spanning vectors (0,1,-1) and (1,-1,-1) of W.
    • Apply A to each spanning vector to get generators of A(W).
    • Check linear independence / span.
    • Extract basis of A(W).
14. 7 Marks
    bFind the linear transformation of $T : IR^2 \to IR^2$ for which $T(2, 3) = (4, 6)$ and $T(1, 0) = (0, 0)$.
    How to answer

    • Use linearity and given images.
    • Express general (x,y) via basis containing (2,3),(1,0) if independent.
    • Note T(1,0)=(0,0) forces dependence.
    • Derive formula for T(x,y).
15. 7 Marks
    a$A = \begin{pmatrix} 4 & 1 & -1 \\ 2 & 5 & -2 \\ 1 & 1 & 2 \end{pmatrix}$ find minimal polynomial of A.
    How to answer

    • Compute characteristic polynomial |A-λI|.
    • Factor to get eigenvalues.
    • Test annihilating polynomials of lower degree.
    • Identify monic polynomial of least degree with p(A)=0.
16. 7 Marks
    b$A = \begin{pmatrix} 1 & 2 & -1 \\ 1 & -1 & 0 \\ 0 & 2 & 1 \end{pmatrix}$ find characteristic equation.
    How to answer

    • Form A-λI for given 3x3 matrix.
    • Compute determinant |A-λI|.
    • Simplify to cubic polynomial.
    • Set =0 for characteristic equation.