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UPSC 2014 Maths Optional Paper 1 Q1a — Step-by-Step Solution

10 marks · Section A

Subspaces · Linear Algebra · asked 6× in 13 yrs · Read the full method →

Question

Find one vector in $\mathbb R^3$ which generates the intersection of $V$ and $W$, where $V$ is the $xy$ plane and $W$ is the space generated by the vectors $(1,2,3)$ and $(1,-1,1)$.

Technique

Parametric description of $W$ + linear constraint from $V$ + back-substitution.

Solution

Strategy. Write a general element of $W$, impose the $V$ condition ($z=0$), solve the resulting linear constraint.

Step 1 — Parametrise $W$

$W=\{a(1,2,3)+b(1,-1,1):a,b\in\mathbb R\}=\{(a+b,\;2a-b,\;3a+b)\}$.

Step 2 — Intersect with $V=\{(x,y,0)\}$

Require $z$-component $=0$:

Step 3 — Substitute

For any scalar $a$:

Hence $V\cap W=\operatorname{span}\{(-2,5,0)\}$ — a one-dimensional subspace.

Answer