Maths 2 Quiz 2: 20 November 2022 (September 2022 term)

Which of the following options is/are true?

Which of the following options is/are true?

Let W be a proper subspace of an inner product space V , where dim(V) = 3 and PW be the projection of V on W. Answer the subquestion based on the given data.

Let W be a proper subspace of an inner product space V , where dim(V) = 3 and PW be the projection of V on W. Answer the subquestion based on the given data.

Which of the following option is/are true?

Which of the following is/are unit vectors in V ?

Consider a linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^2$ such that the matrix representation of $T$ is $A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 2 & 3 \end{bmatrix}$ with respect to the ordered bases $\beta = \{(1,0,0), (0,1,0), (1,1,1)\}$ and $\gamma = \{(1,0), (1,1)\}$ for the domain and codomain, respectively. Answer the subquestions based on the given data.

Nullity of the matrix A is

Consider a linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^2$ such that the matrix representation of $T$ is $A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 2 & 3 \end{bmatrix}$ with respect to the ordered bases $\beta = \{(1,0,0), (0,1,0), (1,1,1)\}$ and $\gamma = \{(1,0), (1,1)\}$ for the domain and codomain, respectively. Answer the subquestions based on the given data.

Which of the following option is true?

Consider a linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^2$ such that the matrix representation of $T$ is $A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 2 & 3 \end{bmatrix}$ with respect to the ordered bases $\beta = \{(1,0,0), (0,1,0), (1,1,1)\}$ and $\gamma = \{(1,0), (1,1)\}$ for the domain and codomain, respectively. Answer the subquestions based on the given data.

The teacher asked Soumya and Sohini to consider an affine space each. Soumya considered the affine subspace $L$ and Sohini considered the affine subspace $L'$ of $\mathbb{R}^3$, where $L = U$ and $L' = (2,0,1) + U'$, for some vector subspaces $U = Span\{(2,0,1), (1,1,0), (0,1,0)\}$ and $U' = Span\{(1,0,1), (0,1,1)\}$ of $\mathbb{R}^3$. Suppose there is a linear transformation $T : U \to U'$ such that $(0,1,0) \in ker(T)$, $T(2,0,1) = (0,1,1)$ and $T(1,1,0) = (1,0,1)$. An affine mapping $f : L \to L'$ is obtained by defining $f(u) = (2, 0, 1) + T(u)$, for all $u \in U$. By using the above information answer the given subquestions:

Which of the following affine subspaces was considered by Soumya?

The teacher asked Soumya and Sohini to consider an affine space each. Soumya considered the affine subspace $L$ and Sohini considered the affine subspace $L'$ of $\mathbb{R}^3$, where $L = U$ and $L' = (2,0,1) + U'$, for some vector subspaces $U = Span\{(2,0,1), (1,1,0), (0,1,0)\}$ and $U' = Span\{(1,0,1), (0,1,1)\}$ of $\mathbb{R}^3$. Suppose there is a linear transformation $T : U \to U'$ such that $(0,1,0) \in ker(T)$, $T(2,0,1) = (0,1,1)$ and $T(1,1,0) = (1,0,1)$. An affine mapping $f : L \to L'$ is obtained by defining $f(u) = (2, 0, 1) + T(u)$, for all $u \in U$. By using the above information answer the given subquestions:

Which of the following affine subspaces was considered by Sohini?

The teacher asked Soumya and Sohini to consider an affine space each. Soumya considered the affine subspace $L$ and Sohini considered the affine subspace $L'$ of $\mathbb{R}^3$, where $L = U$ and $L' = (2,0,1) + U'$, for some vector subspaces $U = Span\{(2,0,1), (1,1,0), (0,1,0)\}$ and $U' = Span\{(1,0,1), (0,1,1)\}$ of $\mathbb{R}^3$. Suppose there is a linear transformation $T : U \to U'$ such that $(0,1,0) \in ker(T)$, $T(2,0,1) = (0,1,1)$ and $T(1,1,0) = (1,0,1)$. An affine mapping $f : L \to L'$ is obtained by defining $f(u) = (2, 0, 1) + T(u)$, for all $u \in U$. By using the above information answer the given subquestions:

Which of the following functions represents f correctly?