Maths 2 Quiz 1: 27 October 2024 (September 2024 term)

Choose the correct option(s) from the following:

Consider the system of linear equations Ax = b. Choose all the options from the following which guarantee a unique solution.

Choose all the correct options from the following.

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

Write correct answers for the given subquestions.

Write correct answers for the given subquestions.

Write correct answers for the given subquestions.

Find the dimension of the subspaces given in the subquestions.

Find the dimension of the subspaces given in the subquestions.

Find the dimension of the subspaces given in the subquestions.

Consider the following sets.

$\mathcal{B}_1 = \left\{\begin{bmatrix} 3 & -3 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ -5 & 5 \end{bmatrix}\right\},$

$\mathcal{B}_2 = \left\{\begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}\right\},$

$\mathcal{B}_3 = \left\{\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \begin{bmatrix} 0 & -2 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 3 & 0 \end{bmatrix}\right\}.$

Choose the correct basis for each of the given vector spaces in the subquestions.

Consider the following sets.

$\mathcal{B}_1 = \left\{\begin{bmatrix} 3 & -3 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ -5 & 5 \end{bmatrix}\right\},$

$\mathcal{B}_2 = \left\{\begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}\right\},$

$\mathcal{B}_3 = \left\{\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \begin{bmatrix} 0 & -2 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 3 & 0 \end{bmatrix}\right\}.$

Choose the correct basis for each of the given vector spaces in the subquestions.

W2 is the set of all 2 × 2 matrices such that the sum of entries in each row is zero.

Consider the following sets.

$\mathcal{B}_1 = \left\{\begin{bmatrix} 3 & -3 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ -5 & 5 \end{bmatrix}\right\},$

$\mathcal{B}_2 = \left\{\begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}\right\},$

$\mathcal{B}_3 = \left\{\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \begin{bmatrix} 0 & -2 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 3 & 0 \end{bmatrix}\right\}.$

Choose the correct basis for each of the given vector spaces in the subquestions.

W3 is the set of all 2 × 2 matrices such that the sum of the diagonal entries is

Answer the given subquestions.

Answer the given subquestions.

Consider the matrix $A = \begin{bmatrix} 1 & 2 & -3 \\ -2 & 1 & -4 \\ 1 & 0 & 1 \end{bmatrix}$. Consider the system of linear equations $Ax = \begin{bmatrix} 3 \\ b \\ 1 \end{bmatrix}$. Find the value of $b$ if the system has infinitely many solutions.