UPSC 2018 Maths Optional Paper 1 Q1b — Step-by-Step Solution

10 marks · Section A

Bases and dimension; coordinates in a basis · Linear Algebra · asked 7× in 13 yrs · Read the full method →

Question

Express basis vectors $e_1=(1,0)$ and $e_2=(0,1)$ as linear combinations of $\alpha_1=(2,-1)$ and $\alpha_2=(1,3)$ .

Coordinates in a new basis via the inverse of the matrix whose columns are the basis vectors.

Write $e_j=c_1\alpha_1+c_2\alpha_2$ . With $\alpha_1=(2,-1),\ \alpha_2=(1,3)$ , the coefficient matrix (columns $\alpha_1,\alpha_2$ ) is

M=\begin{pmatrix}2&1\\-1&3\end{pmatrix},\qquad \det M=2\cdot3-(1)(-1)=7\neq0,

so $\{\alpha_1,\alpha_2\}$ is a basis and the expansions exist and are unique.

M^{-1}=\frac1{\det M}\begin{pmatrix}3&-1\\1&2\end{pmatrix}=\frac17\begin{pmatrix}3&-1\\1&2\end{pmatrix}.

The columns of $M^{-1}$ give the coordinates of $e_1$ and $e_2$ in the basis $\{\alpha_1,\alpha_2\}$ .

e_1=\tfrac37\alpha_1+\tfrac17\alpha_2,\qquad e_2=-\tfrac17\alpha_1+\tfrac27\alpha_2.

\boxed{\;e_1=\tfrac{3}{7}\alpha_1+\tfrac{1}{7}\alpha_2,\qquad e_2=-\tfrac{1}{7}\alpha_1+\tfrac{2}{7}\alpha_2.\;}