IITM BS Question 4 - Week 6 Practice

Prasnya

IITM BS Question 4 - Week 6 Practice | Prasnya

00:00

Q4Comprehension5 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. Define a linear transformation

T: M_{2 \times 2} \to \mathbb{R}

T\left(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\right) = a + b + c + d

. 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}

, and

B = \{1\}

be a basis for

\mathbb{R}

Suppose

n

denotes the nullity of

T

and

m

denotes the dimension of the subspace of

M_{2 \times 2}

spanned by

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

. Then, what is the value of

n - m

Your answer

Press Enter to check.

00:00

21 Dec 2025

Mathematics 2 · Week 6

Q4.Suppose

n

denotes the nullity of

T

and

m

denotes the dimension of the subspace of

M_{2 \times 2}

spanned by

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

. Then, what is the value of

n - m

?@passage

Your Answer

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00:00

Q4Comprehension5 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. Define a linear transformation

T: M_{2 \times 2} \to \mathbb{R}

T\left(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\right) = a + b + c + d

. 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}

, and

B = \{1\}

be a basis for

\mathbb{R}

Suppose

n

denotes the nullity of

T

and

m

denotes the dimension of the subspace of

M_{2 \times 2}

spanned by

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

. Then, what is the value of

n - m

Your answer

Press Enter to check.

00:00

21 Dec 2025

Mathematics 2 · Week 6

Q4.Suppose

n

denotes the nullity of

T

and

m

denotes the dimension of the subspace of

M_{2 \times 2}

spanned by

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

. Then, what is the value of

n - m

?@passage

Your Answer

Save

Read

Check

Details