(a) Let H be a subspace of R⁴ spanned by the vectors v₁ = (1, –2, 5, –3), v₂ = (2, 3, 1, –4), v₃ = (3, 8, –3, –5). Then find a basis and dimension of H, and extend the basis of H to a basis of R⁴. (10 marks)

(b) Let T : R³ → R³ be a linear operator

Question

(a) Let H be a subspace of R⁴ spanned by the vectors v₁ = (1, –2, 5, –3), v₂ = (2, 3, 1, –4), v₃ = (3, 8, –3, –5). Then find a basis and dimension of H, and extend the basis of H to a basis of R⁴. (10 marks)

(b) Let T : R³ → R³ be a linear operator and B = {v₁, v₂, v₃} be a basis of R³ over R. Suppose that Tv₁ = (1, 1, 0), Tv₂ = (1, 0, –1), Tv₃ = (2, 1, –1). Find a basis for the range space and null space of T. (10 marks)

(c) Discuss the continuity of the function

f(x) = { 1/(1–e^(–1/x)), x ≠ 0
       { 0, x = 0

for all values of x. (10 marks)

(d) Expand ln(x) in powers of (x–1) by Taylor's theorem and hence find the value of ln(1·1) correct up to four decimal places. (10 marks)

(e) Find the equation of the right circular cylinder which passes through the circle x² + y² + z² = 9, x – y + z = 3. (10 marks)

UPSC Answer Check · Accepted Answer

Solve each sub-part systematically with approximately equal time allocation (2 marks per minute). For (a), use row reduction to find basis and extend; for (b), construct matrix representation to find range and null spaces; for (c), analyze left and right limits at x=0 and behavior at other points; for (d), apply Taylor expansion about x=1 with Lagrange remainder; for (e), use the standard cylinder formula with given circle as directrix. Present solutions clearly with proper mathematical notation.
- For (a): Correctly row reduce the matrix formed by v₁, v₂, v₃ to identify linearly independent vectors, state dim(H)=2, and find two appropriate vectors to extend basis to R⁴
- For (b): Construct the matrix of T relative to standard basis, correctly identify rank(T)=2 for range space basis, and solve Tx=0 to find null space basis with dimension 1
- For (c): Evaluate lim(x→0⁺) f(x) = 1, lim(x→0⁻) f(x) = 0, conclude discontinuity at x=0 (jump discontinuity), and verify continuity elsewhere
- For (d): Derive Taylor series ln(x) = Σ(-1)^(n+1)(x-1)^n/n for 0<x≤2, compute ln(1.1) ≈ 0.0953 using first 4-5 terms with error estimation
- For (e): Identify center (0,0,0), radius √3, axis direction (1,-1,1), and derive cylinder equation (x+y)² + (y+z)² + (z+x)² = 18 or equivalent standard form

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly sets up augmented matrix for (a), matrix representation of T for (b), proper limit analysis framework for (c), Taylor expansion about correct point for (d), and identifies cylinder parameters (center, radius, axis) for (e)	Sets up most problems correctly but has minor errors in matrix dimensions, wrong expansion point, or incomplete parameter identification	Major setup errors: wrong matrix size, incorrect basis identification, wrong center/radius for cylinder, or fundamentally flawed approach
Method choice	20%	10	Uses Gaussian elimination for (a), matrix method over change-of-basis for (b), systematic ε-δ or sequential limit approach for (c), Taylor with remainder term for (d), and standard cylinder equation derivation for (e)	Uses correct general methods but inefficient alternatives or misses optimal approaches like using rank-nullity theorem directly	Inappropriate methods: solving by inspection only, ignoring matrix representation, or using numerical approximation where exact analysis is required
Computation accuracy	20%	10	Flawless arithmetic: correct row reduction with precise pivot operations, accurate determinant/rank calculations, correct limit values (e^(-∞)=0, e^(+∞)=∞), Taylor coefficients 1, -1/2, 1/3..., and exact cylinder equation	Minor computational slips: sign errors in row operations, one incorrect limit value, or arithmetic error in final decimal place of ln(1.1)	Major computational errors: incorrect rank, wrong null space vectors, reversed limits, or completely wrong cylinder equation
Step justification	20%	10	Explicitly justifies: why v₁,v₂ are independent in (a); rank-nullity application in (b); one-sided limit analysis with proper exponential behavior in (c); convergence interval and error bound in (d); why axis is perpendicular to plane in (e)	Shows key steps with minimal justification, states results without proving linear independence or verifying limit existence	Unjustified leaps: claims basis without verification, asserts continuity without limits, or presents final answers without derivation
Final answer & units	20%	10	Clear final answers: explicit basis vectors with dimension for (a); explicit basis for range and null space with dimensions for (b); precise continuity classification for (c); ln(1.1)=0.0953 with error bound for (d); simplified standard cylinder equation for (e)	Correct answers but poorly formatted: missing dimension statements, unverified decimal approximation, or unsimplified equation	Missing or wrong final answers: incorrect basis, wrong continuity conclusion, or no equation for the cylinder

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