Mathematics for Data Science II End Term: 31 August 2025, Set QDF1 (May 2025 term)

Let $T_1, T_2 : \mathbb{R}^2 \to \mathbb{R}^2$ be two linear transformations defined as:

$T_1(x,y) = (3x - y, 2x - y), \quad T_2(x,y) = (2x + \alpha y, \alpha x)$

Let $A$ and $B$ be the matrix representations of $T_1$ and $T_2$ with respect to the standard bases of $\mathbb{R}^2$, respectively.

Consider the following statements:

S1: Considering the basis $\{e_1 + e_2, -e_2\}$, we can see that if $\alpha = 1$, $A$ and $B$ are similar matrices.

S2: If $\alpha = 2$, $A$ and $B$ are similar matrices.

Which of the following options is true?

Which of the following options are correct?
Let A and B be square matrices of the same order.

Let $f : \mathbb{R}^2 \to \mathbb{R}$ be a function for which the following is known.

• At every point, the partial derivatives of $f$ exist and are continuous.
• $f(1,-1) = 2$
• The tangent line to the graph of $f$ at $(1,-1)$ in the direction $(1,0)$ contains the point $(2,-1,6)$.
• The tangent line to the graph of $f$ at $(1,-1)$ in the direction $(0,1)$ contains the point $(1,-2,2)$.

Find the directional derivative of $f$ at $(1,-1)$ in the direction $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

Based on the above data, answer the given subquestions.

Find the value of k.

Based on the above data, answer the given subquestions.

Find the dimension of the affine subspace S.

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

If (a, 3,−2) is a vector which lies along a direction in which there is no change in f, then find the value of a.

Based on the above data, answer the given subquestions.

Let S denote the set of all unit directions along which there is no change in f. Find dim(Span(S)).

Based on the above data, answer the given subquestions.

If the gradient of f at (1,−1, 1) is given by (12, 4, c), then find the value of c.

Based on the above data, answer the given subquestions.

Which of the following points lie on T(1,−1)?

Based on the above data, answer the given subquestions.

Let W denote the subspace corresponding to the affine subspace T(1,−1). Which of the following is a basis for the orthogonal complement of W?

The “best approximate solution” to an inconsistent system of linear equations $A\underset{\sim}{x} = \underset{\sim}{b}$ can be found by calculating solutions to $A\underset{\sim}{x} = \underset{\sim}{b}'$ where $\underset{\sim}{b}'$ is the projection of the vector $\underset{\sim}{b}$ onto the column space of $A$. Note that $\underset{\sim}{b}'$ is also the vector in the column space of $A$ at minimum distance from $\underset{\sim}{b}$.

Consider the system of equations given by $A\underset{\sim}{x} = \underset{\sim}{b}$, where

$A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & -1 & 0 \\ 1 & 0 & 1 \end{bmatrix}, \quad \underset{\sim}{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \quad \underset{\sim}{b} = \begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix}.$

It can be checked that the above system of linear equations is inconsistent. Answer the given subquestions to find the best approximate solution for the given system of equations.

The set $\{(1,-1,0),(2,0,1)\}$ is a basis for the column space of the coefficient matrix $A$. Thus, any vector in the column space of $A$ is given by $x(1,-1,0) + y(2,0,1)$ (we use row vector representation for convenience). Using this representation for vectors in the column space of $A$, which of the following is the expression of the function $\phi : \mathbb{R}^2 \to \mathbb{R}$ which measures the square of the distance between the given vector in the column space of $A$, and $\underset{\sim}{b}$.

The “best approximate solution” to an inconsistent system of linear equations $A\underset{\sim}{x} = \underset{\sim}{b}$ can be found by calculating solutions to $A\underset{\sim}{x} = \underset{\sim}{b}'$ where $\underset{\sim}{b}'$ is the projection of the vector $\underset{\sim}{b}$ onto the column space of $A$. Note that $\underset{\sim}{b}'$ is also the vector in the column space of $A$ at minimum distance from $\underset{\sim}{b}$.

Consider the system of equations given by $A\underset{\sim}{x} = \underset{\sim}{b}$, where

$A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & -1 & 0 \\ 1 & 0 & 1 \end{bmatrix}, \quad \underset{\sim}{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \quad \underset{\sim}{b} = \begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix}.$

It can be checked that the above system of linear equations is inconsistent. Answer the given subquestions to find the best approximate solution for the given system of equations.

Which of the following is the only critical point for the function defined in the previous part?

The “best approximate solution” to an inconsistent system of linear equations $A\underset{\sim}{x} = \underset{\sim}{b}$ can be found by calculating solutions to $A\underset{\sim}{x} = \underset{\sim}{b}'$ where $\underset{\sim}{b}'$ is the projection of the vector $\underset{\sim}{b}$ onto the column space of $A$. Note that $\underset{\sim}{b}'$ is also the vector in the column space of $A$ at minimum distance from $\underset{\sim}{b}$.

Consider the system of equations given by $A\underset{\sim}{x} = \underset{\sim}{b}$, where

$A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & -1 & 0 \\ 1 & 0 & 1 \end{bmatrix}, \quad \underset{\sim}{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \quad \underset{\sim}{b} = \begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix}.$

It can be checked that the above system of linear equations is inconsistent. Answer the given subquestions to find the best approximate solution for the given system of equations.

What can be concluded from the Hessian test at the only critical point for ϕ?

The “best approximate solution” to an inconsistent system of linear equations $A\underset{\sim}{x} = \underset{\sim}{b}$ can be found by calculating solutions to $A\underset{\sim}{x} = \underset{\sim}{b}'$ where $\underset{\sim}{b}'$ is the projection of the vector $\underset{\sim}{b}$ onto the column space of $A$. Note that $\underset{\sim}{b}'$ is also the vector in the column space of $A$ at minimum distance from $\underset{\sim}{b}$.

Consider the system of equations given by $A\underset{\sim}{x} = \underset{\sim}{b}$, where

$A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & -1 & 0 \\ 1 & 0 & 1 \end{bmatrix}, \quad \underset{\sim}{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \quad \underset{\sim}{b} = \begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix}.$

It can be checked that the above system of linear equations is inconsistent. Answer the given subquestions to find the best approximate solution for the given system of equations.