Mathematics for Data Science: what's likely to come

Let $V$ be a subspace of $\mathbf{R}^n$ and $w$ a vector in $\mathbf{R}^n$. a) If $\{v_1, v_2, \ldots, v_k\}$ be an orthogonal basis for $V$, then

is the projection of $w$ onto $V$. b) Moreover, if $\{v_1, v_2, \ldots, v_k\}$ is an orthonormal basis for $V$, then

is the projection of $w$ onto $V$.

OR

Consider the vectors $u = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $v = \begin{pmatrix} -1 \\ 1 \end{pmatrix}$. a) Write the vector $w = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$ in terms of the vectors $u$ and $v$. b) Show that the vectors $u$ and $v$ span $\mathbf{R}^2$.

Define the span of a set. Prove that span $\{v_1, v_2, \ldots, v_k\} \subseteq V$ is a subspace of a vector space $V$. Also, show that if $x \in \mathbf{R}^2$, such that $x \neq 0$, then $x^{\perp}$ is a subspace of $\mathbf{R}^2$.

Let $u_1 = (1, 2, 2, -1)$, $u_2 = (1, 1, -1, 1)$, $u_3 = (-1, 1, -1, -1)$ and $B = \{u_1, u_2, u_3\}$ an orthogonal basis for $V = \text{span} \{u_1, u_2, u_3\}$. Find the projection of $w = (0, 1, 2, 3)$ onto $V$.

Define linearly independent vectors. Prove that an orthogonal set of nonzero vectors in a vector space is linearly independent.

What is the angle between the diagonal of the unit cube in the positive orthant and the vector $e_1$ in $\mathbf{R}^3$?

Show that

a. The line $x_2 = \alpha x_1$ is a subspace $\mathbb{R}^2$. b. The set of points that is the union of two lines through the origin is not a subspace.

Let

Show that $B = \{v_1, v_2\}$ is an orthonormal basis for $\mathbb{R}^2$. Find a vector $x \in \mathbb{R}^2$ with respect to the basis $B$.

Show that $u = \begin{pmatrix} -1 \\ 1 \end{pmatrix}$ and $v = \begin{pmatrix} -1 \\ -1 \end{pmatrix}$ are orthogonal in $\mathbf{R}^2$ and find corresponding orthonormal basis for $\mathbf{R}^2$.

(a) Let $S : \mathbf{R}^t \to \mathbf{R}^m$ and $T : \mathbf{R}^n \to \mathbf{R}^t$ be linear transformations, given by matrices $A$ and $B$, respectively. Prove that the composition $S \circ T : \mathbf{R}^n \to \mathbf{R}^m$ is a linear transformation and is given by $AB$. (b) Let $x = x_1 e_1 + x_2 e_2$, where $e_1$ and $e_2$ are unit basis vectors of $\mathbf{R}^2$. When does the equality $x = 0$ hold? What does $x = 0$ mean?

Explain four major ways to view a matrix. Prove that if $S : \mathbf{R}^l \to \mathbf{R}^m$ and $T : \mathbf{R}^n \to \mathbf{R}^l$ are linear transformations, given by matrices $A$ and $B$, respectively, then, the composition $S \circ T : \mathbf{R}^n \to \mathbf{R}^m$ is a linear transformation and is given by $AB$.

OR

Let $S$ and $T$ be matrix transformations defined by $S(v) = Ay$ and $T(x) = Bx$, where

a) What are the domains and codomains of $S$ and $T$? Why is the composite transformation $S \circ T$ defined? What are the domain and the codomain of $S \circ T$? b) Let $x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$. Determine $T(x)$. c) Find $(S \circ T)(x)$. d) Find a matrix $C$ so that $(S \circ T)(x) = Cx$. e) Show that $S \circ T$ is linear.

Let $S$ and $T$ be matrix transformations defined by $S(y) = Ay$ and $T(x) = Bx$, where

a) What are the domains and codomains of $S$ and $T$? Why is the composite transformation $S \circ T$ defined? What are the domain and the codomain of $S \circ T$? b) Let $x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$. Determine $T(x)$. c) Find $(S \circ T)(x)$. d) Find a matrix $C$ so that $(S \circ T)(x) = Cx$. e) Show that $S \circ T$ is linear.

Prove that if $\theta$ is the angle between two non-zero vectors $x$, $y$ in $\mathbf{R}^n$, then

a) Let $A$ be an $m \times n$ matrix, $B$ an $n \times p$ matrix, and $C$ a $p \times q$ matrix, so that $(AB)C$ and $A(BC)$ are defined. Prove that $(AB)C = A(BC)$. Determine the size of the matrix $A(BC)$.

b) Let $x = x_1 e_1 + x_2 e_2$, where $e_1$ and $e_2$ are unit basis vectors of $\mathbf{R}^2$. When does the equality $x = 0$ hold? What does $x = 0$ mean?

Let $V$ be a subspace of $\mathbf{R}^n$ and $w$ a vector in $\mathbf{R}^n$.

a) If $\{v_1, v_2, \ldots, v_k\}$ is an orthogonal basis for $V$, then prove that

is the projection of $w$ onto $V$.

b) Moreover, if $\{v_1, v_2, \ldots, v_k\}$ is an orthonormal basis for a vector space $V$, then prove that

is the projection of $w$ onto $V$.

OR

Give a geometric description of $\text{span}(v)$ and $\text{span}(u, v)$. Consider the vectors $u = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $v = \begin{pmatrix} -1 \\ 1 \end{pmatrix}$.

a) Write the vector $w = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$ in terms of the vectors $u$ and $v$. b) Show that the vectors $u$ and $v$ span $\mathbf{R}^2$.

Let

Show that $V$ is a subspace of $\mathbf{R}^3$, and $B$ is a basis for $V$.

OR

Show that the following set of vectors is a basis for $\mathbf{R}^3$, and then express the standard basis vectors: $e_1$, $e_2$, $e_3$ in terms of these:

Let

Show that $V$ is a subspace of $\mathbf{R}^3$, and $B$ is a basis for $V$.

OR

Let $V$ be a subspace of $\mathbf{R}^n$ and $w$ a vector in $\mathbf{R}^n$. a) If $\{v_1, v_2, \ldots, v_k\}$ is an orthogonal basis for $V$, derive the expression for the projection of $w$ onto $V$. b) If $\{v_1, v_2, \ldots, v_k\}$ is an orthonormal basis for $V$, derive the expression for the projection of $w$ onto $V$.

What is $L_\infty$ norm on a vector $n$-space $R^n$? Write any two properties. Let $\|.\|$ be the Euclidean norm, $X$ and $Y$ be two vectors in $R^n$. State and prove the Triangle Inequality and Parallelogram Law. Verify Cauchy-Schwarz inequality for $X = (1, 3)$ and $Y = (2, 1)$. (1 + 2 + 2 + 1)

Define Kernel and Image of a linear transformation $T : V \to W$. If $v_1, v_2, \ldots, v_n$ are linearly independent vectors in $V$ and $\operatorname{Ker} T = \{0_V\}$. Is the set $\{ T(v_1), T(v_2), \ldots, T(v_n) \}$ forms linearly independent vectors in $W$? Justify. Also, show that the set $\{(0, 1), (1, 1)\}$ of vectors span $R^2$. (1 + 2.5 + 2.5)

OR

What is quadratic form? Let $A$ be a $3 \times 3$ square matrix with a quadratic form in 3 variables. Then there exists a $3 \times 3$ symmetric matrix $B$ such that $X^T A X = X^T B X, \forall X \in R^3$. Further, express the quadratic form $x_1^2 + x_1 x_2 - 4 x_3 x_1 + 2 x_2 x_3 - 4 x_3^2$ as the difference of squares. (1 + 2.5 + 2.5)

Define $L_1, L_2$ and $L_\infty$ norms on a vector $n$-space $R^n$. Let $\|.\|$ be the Euclidean norm, and $X$ & $Y$ be two vectors in $R^n$. Prove the Cauchy-Schwarz inequality $|X.Y| \leq \|X\| \|Y\|$. Verify this property for $X = (1, 3)$ and $Y = (2, 1)$. (1+4+1)

Distinguish between linear dependent and independent vectors. Prove that the linear hull of a given set of vectors $v_1, v_2, \ldots, v_n$ in a vector space $V$ is a subspace of $V$. Also, the representation of any vector in a vector space in terms of its basis vectors is unique. (1+2.5+2.5)

OR

Define a basis and dimension of a vector space. Show that the set $B = \{(1, 1, 1), (1, -1, 1), (2, 0, 3)\}$ forms a basis for $R^3$. Also, find the co-ordinates of $v = (1, 3, 2) \in R^3$ with respect to the basis $B$. (1+3+2)

What are the conditions necessary for a matrix to possess an inverse? Prove that the inverse of a square matrix if it exists, is unique. If $A$, $B$, and $C$ are matrices of order $m \times n$, $n \times p$, $p \times q$ respectively, then prove that $(AB)C = A(BC)$. (1+1+4)

OR

Define quadratic form. Let $A$ be a $3 \times 3$ square matrix with a quadratic form in 3 variables. Then there exists a $3 \times 3$ symmetric matrix $B$ such that $X^T A X = X^T B X$, $\forall X \in R^3$. Further, express the quadratic form $x_1^2 + x_1 x_2 - 4 x_3 x_1 + 2 x_2 x_3 - 4 x_3^2$ as the difference of squares. (1+2.5+2.5)

Let $u_1 = (2\ 0\ 0)^T$, $u_2 = (0\ 1\ 1)^T$ and $u_3 = (0\ 1\ -1)^T$. Find the orthonormal set associated with the set $S = \{u_1, u_2, u_3\}$.

Prove that an orthogonal set of nonzero vectors in a vector space is linearly independent.

(a) By showing that the $L_\infty$-norm satisfies each of the conditions in the definition of a norm prove this is a vector norm for $\mathbb{R}^n$.

(b) Let $x = (x_1, \ldots, x_n) \in \mathbb{R}^n$ be a vector with $x_i = i^{-1}$. Compute the 1-norm, the 2-norm, and the $\infty$-norm of $x$.

OR

Prove that if $x$ and $y$ are vectors in $\mathbb{R}^n$, then

(a) $|x \cdot y| \leq \|x\|_2 \|y\|_2$.

Explain four major ways to view a matrix. Prove that if $S : \mathbb{R}^l \to \mathbb{R}^m$ and $T : \mathbb{R}^n \to \mathbb{R}^l$ are linear transformations, given by matrices $A$ and $B$, respectively, then, the composition $S \circ T : \mathbb{R}^n \to \mathbb{R}^m$ is a linear transformation and is given by $AB$.

By showing that the $L_1$-norm satisfies each of the conditions in the definition of a norm prove this is a vector norm. First do this for $\mathbb{R}^2$, and then do this for $\mathbb{R}^n$.

OR

Prove that if $x \in \mathbb{R}^n$, then $\|x\|_\infty \leq \|x\|_1 \leq n\|x\|_\infty$.

Let

Show that $B$ is a basis for $V$.

OR

Let $V$ be a subspace of $\mathbb{R}^n$ and $w$ a vector in $\mathbb{R}^n$.

(a) If $\{v_1, v_2, \ldots, v_k\}$ is an orthogonal basis for $V$, derive the expression for the projection of $w$ onto $V$.

(b) If $\{v_1, v_2, \ldots, v_k\}$ is an orthonormal basis for $V$, derive the expression for the projection of $w$ onto $V$.

Write the vector $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ in terms of the vectors $u = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $v = \begin{pmatrix} -1 \\ 1 \end{pmatrix}$.

Define $L_1$, $L_2$ and $L_\infty$ norms on $\mathbf{R}^n$. Calculate $L_1$, $L_2$ and $L_\infty$ norms of the vector $x = (1, -1, 0, \ldots, 0, 2)$ on $\mathbf{R}^n$.

Show that

a) The line $x_2 = ax_1$ is a subspace $\mathbf{R}^2$. b) The line $x_2 = ax_1 + b$ is not a subspace $\mathbf{R}^2$ for $b \neq 0$.

Are the following sets form the subspace of $R^2$? Justify.

a) The set $S = \left\{ \begin{pmatrix} x \\ y \\ z \end{pmatrix} : y + z = 2 \text{ and } x, y, z \in R \right\}$ in the vector space $R^3$.

b) The closed $L_2$ ball, $B(0,3) = \left\{ X = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \in R^2 : \|X\| \le 9 \right\}$. (1.5 + 1.5)

Let $T : R^2 \to R^2$ be defined by $T(1, 0) = (1, 3)$, $T(1, 2) = (0, -2)$. If $T$ is linear, find the formula for $T(x, y)$. What is the matrix represented by $T$ relative to the standard basis? (1.5 + 1.5)

Are the following sets form the subspace of $R^2$? Justify.

a) The set $S$ of all solutions of homogeneous equation $AX = 0$ of any $2 \times 2$ matrix $A$ and $X = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \in R^2$.

b) The closed $L_2$ ball $= B(0,1) = \{ X = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \in R^2 : \|X\| \leq 1 \}$. (1.5+1.5)

Define an involutory matrix. Is the matrix $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ involutory? Justify. Prove that the inverse of the transpose of a non-singular matrix $A$ is the transpose of its inverse. (0.5+1+1.5)

Define linear transformation. Let $T: R^2 \to R^2$ be defined by $T(0, 1) = (2, 1)$, $T(1, 4) = (0, -2)$. If $T$ is linear, find the formula for $T(x, y)$. Also, find the matrix represented by $T$ relative to the standard basis. (0.5+1.5+1)

What is meant by linear independence of a set of vectors. Let $V$ and $W$ be two vector spaces over the field $F$ and $T : V \to W$ be a linear transformation. Then prove that the kernel of $T$ is a subspace of $V$ and the image of $T$ is a subspace of $W$. (1 + 1 + 1)

Show that any vector in $\mathbb{R}^3$ can be expressed as a linear combination of the three unit basis vectors in $\mathbb{R}^3$. Also, show that a linear combination of the three unit basis vectors in $\mathbb{R}^3$ equals to 0 if and only if all coefficients in the linear combination are zeros.

How many kinds of subspaces of $\mathbb{R}^3$ are there? Mention them. Show that a plane through the origin is a two-dimensional subspace of $\mathbb{R}^3$.