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MATH 101: Assignment 01

DETAILS
Total Marks 100
Instructions
  • This assignment must be submitted handwritten. Typed submissions will NOT be accepted. I want to see your handwriting!
  • You may choose which questions to attempt as specified in each section.
  • Show your thinking clearly. Use words, and be verbose about your thoughts. Answers without reasoning will receive no marks. At the same time, I don't want you to drag the answers. One clear argument wins over thousand confusing ones.
  • If your score crosses 100 by virtue of bonus problems, the extra marks will get compensated in the next assignment.
Duration 1 week
Late Submission Submission delayed by each day will incur a penalty of \(5\) marks per day.

Section A: Solving Linear Systems & Elimination (20 Marks)

Attempt any 2 out of the following 3 problems. Each problem carries 10 marks.

A1. Elimination and Comparison of Systems

Use Gaussian elimination to solve the following system of equations.

\[ \begin{aligned} u + v + w &= 6 \\ u + 2v + 2w &= 11 \\ 2u + 3v - 4w &= 3 \end{aligned} \]

and

\[ \begin{aligned} u + v + w &= 7 \\ u + 2v + 2w &= 10 \\ 2u + 3v - 4w &= 3 \end{aligned} \]

Compare the two systems and comment on the nature of their solutions.

A2. Elimination and Pivot Analysis

Use Gaussian elimination to solve the following system.

\[ \begin{aligned} x + y + z &= 5 \\ 2x + 3y + 4z &= 14 \\ 3x + 4y + kz &= 18 \end{aligned} \]
  1. Solve the system for a general value of \(k\).
  2. Determine the value(s) of \(k\) for which the system has a unique solution and no solution.

A3. Singularity, Infinite Solutions, and Particular Solutions

Which value of \(q\) makes the following system singular? For that value of \(q\), which right-hand side \(t\) gives infinitely many solutions?

\[ \begin{aligned} x + 4y - 2z &= 1 \\ x + 7y - 6z &= 6 \\ 3y + qz &= t \end{aligned} \]

For the infinite-solution case, find the solution satisfying \(z = 1\).

Section B: Consistency, Column Space, and Solvability (10 Marks)

Attempt 1 problem.

B1. Conditions for Solvability

Under what condition(s) on \(b_1, b_2, b_3\) is the system solvable?

\[ \begin{aligned} x + 2y - 2z &= b_1 \\ 2x + 5y - 4z &= b_2 \\ 4x + 9y - 8z &= b_3 \end{aligned} \]

Find all solutions when the system is solvable.

B2. Conditions for Solvability - 2

Consider the system.

\[ \begin{aligned} x - y + z &= b_1 \\ 2x - 2y + 2z &= b_2 \\ x + y - z &= b_3 \end{aligned} \]
  1. Write the augmented matrix \([A \mid b]\).
  2. Find the condition(s) on \(b_1, b_2, b_3\) for which the system is consistent.
  3. Describe all solutions when the condition holds.

Section C: Linear Independence & Dependence (20 Marks)

Attempt any 2 out of the following 3 problems. Each problem carries 10 marks.

C1. Dependence of Vector Differences

Let \(w_1, w_2, w_3\) be linearly independent vectors.

Define,

\[ v_1 = w_2 - w_3,\quad v_2 = w_1 - w_3,\quad v_3 = w_1 - w_2 \]

Show that \(v_1, v_2, v_3\) are linearly dependent, and find a non-trivial linear combination that equals zero.

C2. Independence of Vector Sums

Let \(w_1, w_2, w_3\) be linearly independent vectors.

Define,

\[ v_1 = w_2 + w_3,\quad v_2 = w_1 + w_3,\quad v_3 = w_1 + w_2 \]

Show that \(v_1, v_2, v_3\) are linearly independent.

C3. True or False (Justify Carefully)

State whether each statement is true or false, and give a justification or counterexample.

  1. If vectors \(x_1, \dots, x_m\) span a subspace \(S\), then \(\dim S = m\).
  2. The intersection of two subspaces cannot be empty.
  3. If \(Ax = Ay\), then \(x = y\).
  4. The column space of a matrix has a unique basis obtained from echelon form.
  5. If a square matrix \(A\) has independent columns, then \(A^2\) also has independent columns.

Section D: Linear Transformations & Matrices (10 Marks)

Attempt 1 problem.

D1. Standard Transformations

  1. What matrix represents: (a) Rotation by \(90^\circ\) counterclockwise followed by projection onto the \(x\)-axis? (b) Projection onto the \(x\)-axis followed by projection onto the \(y\)-axis?

  2. The matrix \(A = \begin{bmatrix}1 & 0 \\ 3 & 1\end{bmatrix}\) produces a shearing transformation. Describe its effect on the \(x\)-axis by computing: \(A(1,0), A(2,0), A(-1,0)\).

D2. Composition of Transformations

  1. Find the matrix that represents reflection across the \(x\)-axis, followed by projection onto the line \(y = x\).

  2. Apply this transformation to the vector \((2,1)\), and interpret the result geometrically.

Section E: Determinants (20 Marks)

Attempt any 2 out of the following 3 problems. Each problem carries 10 marks.

E1. Vandermonde Determinant

Prove that

\[ \det \begin{bmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{bmatrix} = (b-a)(c-a)(c-b) \]

E2. Determinants with Parameter

Compute the determinant of the following matrix.

\[ \begin{bmatrix} 1 & t & t^2 & t^3 \\ t & 1 & t & t^2 \\ t^2 & t & 1 & t \\ t^3 & t^2 & t & 1 \end{bmatrix} \]

E3. Determinants and Singularity

Let

\[ A = \begin{bmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & \lambda \end{bmatrix} \]
  1. Compute \(\det(A)\) as a function of \(\lambda\).
  2. For which value(s) of \(\lambda\) is \(A\) singular?
  3. Explain what singularity means in terms of the column space of \(A\).

Section F: Orthogonality, Projections & Inequalities (20 Marks)

Attempt any 2 out of the following 3 problems. Each problem carries 10 marks.

F1. Orthogonality

Which pairs among the following vectors are orthogonal?

\[ v_1 = \begin{bmatrix}1\\2\\-2\\1\end{bmatrix},\quad v_2 = \begin{bmatrix}4\\0\\4\\0\end{bmatrix},\quad v_3 = \begin{bmatrix}1\\-1\\-1\\-1\end{bmatrix},\quad v_4 = \begin{bmatrix}1\\1\\1\\1\end{bmatrix} \]

F2. Schwarz Inequality and Means

Let \(x, y > 0\).

Define

\[ a = (\sqrt{y}, \sqrt{x}), \quad b = (\sqrt{x}, \sqrt{y}) \]

Use the Schwarz inequality to compare,

\[ \frac{x+y}{2} \quad \text{and} \quad \sqrt{xy} \]

F3. Projection and Inner Products

Let \(a = (1,2,2)\) and let \(P\) be the projection matrix onto the line spanned by \(a\).

  1. Compute the projection matrix \(P\).
  2. For vectors \(x = (2,1,0)\) and \(y = (1,0,1)\), verify \(\langle Px, y \rangle = \langle x, Py \rangle\).
  3. Find the angle between \(Px\) and \(Py\).

Bonus Questions (Optional)

X1. Rank-One Update Determinant (Up to +5 Marks)

Let \(M\) be a \(4 \times 4\) matrix with all rows equal to \((a,b,c,d)\).

Compute:

\[ \det(I + M) = \begin{vmatrix} 1+a & b & c & d \\ a & 1+b & c & d \\ a & b & 1+c & d \\ a & b & c & 1+d \end{vmatrix} \]

Partial credit for solving the case \(a=b=c=d=1\).

X2. Rank-One Updates, Geometry, and Invertibility (Up to +10 Marks)

Let \(A\) be an invertible \(n \times n\) matrix, and let \(u, v \in \mathbb{R}^n\) be nonzero vectors. Define the matrix \(B = A + u v^T\).

  1. Show that for any vector \(x\), \(Bx = Ax + u(v^T x)\). Explain geometrically how \(B\) modifies the action of \(A\).

  2. Prove that if \(1 + v^T A^{-1} u \neq 0\), then \(B\) is invertible, and derive an explicit formula for \(B^{-1}\) in terms of \(A^{-1}, u,\) and \(v\).

  3. Show that if \(1 + v^T A^{-1} u = 0\), then the system \(Bx = 0\) has a nonzero solution, and explicitly construct such a vector.

  4. Interpret the condition \(v^T A^{-1} u = -1\) in terms of projection onto a line and explain why it causes a loss of invertibility.