IITM BS Question 6 - Week 6 Practice

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IITM BS Question 6 - Week 6 Practice | Prasnya

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Q6Comprehension5 Marks21 Dec 2025

Passage

Based on the above data, answer the given subquestions. Consider the vector space

M_{2 \times 2}

, the set of all

2 \times 2

matrices with real entries, with the usual addition and scalar multiplication. Let

B_1 = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \right\}

and

B_2 = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} -1 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & -1 \\ 0 & 1 \end{bmatrix} \right\}

be two ordered bases of

M_{2 \times 2}

Define a linear transformation S:

\mathbb{R}^2

-> M_{2 x 2} by S(a, b) =

\begin{bmatrix} a & a \\ b & b \end{bmatrix}

. Consider the following statements. 1) S o T is an isomorphism. 2) T o S is an isomorphism. 3) S o T is injective but not surjective. Let n denote the number of correct statements and n = 0, if no statement is correct. What is n?

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21 Dec 2025

Mathematics 2 · Week 6

Q6.Define a linear transformation S:

\mathbb{R}^2

-> M_{2 x 2} by S(a, b) =

\begin{bmatrix} a & a \\ b & b \end{bmatrix}

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