Calculus & Linear Algebra Unit 1 Quiz

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Questions and Answers

What is the determinant of the matrix ( \begin{bmatrix} 2 & -2 \ 4 & -3 \end{bmatrix} )?

  • 2
  • -2 (correct)
  • 3
  • 11

Which of the following statements is NOT a property of determinants?

  • The determinant is zero if any two rows are identical.
  • The determinant is unchanged when a row is added to another row. (correct)
  • The determinant of a product is the product of the determinants.
  • The determinant changes sign when two rows are swapped.

Which theorem states that every square matrix satisfies its own characteristic polynomial?

  • Cayley-Hamilton Theorem (correct)
  • Cauchy-Schwarz Theorem
  • Lagrange's Theorem
  • Rouché's Theorem

What is the characteristic polynomial of the matrix ( A = \begin{bmatrix} 1 & 2 \ 2 & 1 \end{bmatrix} )?

<p>( x^2 - 3x + 1 ) (C)</p> Signup and view all the answers

If the rank of matrix A is less than its number of columns, what can be concluded?

<p>The matrix has infinitely many solutions. (C)</p> Signup and view all the answers

What are Eigenvalues?

<p>Values such that ( Av = \lambda v ) for some vector v. (B)</p> Signup and view all the answers

What is the sum of the Eigenvalues of the matrix ( \begin{bmatrix} 2 & -2 \ 4 & -3 \end{bmatrix} )?

<p>5 (C)</p> Signup and view all the answers

What represents a major application of matrices in real life?

<p>Weather forecasting (A)</p> Signup and view all the answers

Which statement correctly defines the concept of rank in linear algebra?

<p>Rank is the dimension of the row space of a matrix. (C)</p> Signup and view all the answers

Which of the following statements describes a vector subspace?

<p>A vector subspace must contain the zero vector. (D)</p> Signup and view all the answers

What does linear independence of a set of vectors imply?

<p>No vector can be expressed as a linear combination of the others. (A)</p> Signup and view all the answers

Which of the following equations defines a Linear Transformation?

<p>It preserves vector addition and scalar multiplication. (A)</p> Signup and view all the answers

In the context of Cayley-Hamilton theorem, what does the theorem state about a square matrix?

<p>The matrix satisfies its own characteristic equation. (C)</p> Signup and view all the answers

Which of the following best describes the dimension of a vector space?

<p>It is the number of vectors in the basis. (D)</p> Signup and view all the answers

For a given linear system, what does the term homogeneous refer to?

<p>All equations equal zero. (D)</p> Signup and view all the answers

If a matrix has a nullity of 3, what can be inferred about its rank?

<p>The total number of columns exceeds the rank by 3. (D)</p> Signup and view all the answers

What does Sylvester's Law (Rank and Nullity Theorem) describe?

<p>The rank, dimension, and nullity of a linear transformation. (A)</p> Signup and view all the answers

Is the set S = {(1, 1), (1, -1)} a basis of R²?

<p>No, because they do not span the space. (B)</p> Signup and view all the answers

Which condition must be satisfied for the union of two subspaces U and V to be a subspace of a vector space?

<p>U and V must have a non-empty intersection. (C)</p> Signup and view all the answers

If V is a vector space over a field F, what can be stated about the set Fv which consists of all scalar multiples of a vector v?

<p>Fv is a subspace of V. (B)</p> Signup and view all the answers

Which of the following describes a linear combination of vectors?

<p>A sum of scalar multiples of the vectors. (A)</p> Signup and view all the answers

What is the dimension of the vector space whose basis is { (2,4,6), (0,5,6), (0, 6,6)}?

<p>2, because one vector is a linear combination of the others. (A)</p> Signup and view all the answers

In a vector space V3, which set of vectors is likely to be linearly independent?

<p>{(1, 2, 1), (3, 1, 5), (3, -4, 7)} (B)</p> Signup and view all the answers

Which of the following statements is true regarding the mapping t: R³ -> R³ defined by t(a, b, c) = (a, b, 0)?

<p>It is a linear transformation. (B)</p> Signup and view all the answers

Flashcards

Vector Space Definition

A set of vectors with operations of addition and scalar multiplication, satisfying specific properties (closure, associativity, commutativity, distributivity, identity, inverse).

Vector Subspace Definition

A subset of a vector space that is also a vector space under the same operations, it must contain the zero vector.

Linear Combination of Vectors

A vector formed by multiplying vectors by scalars and adding the results.

Linear Span of Vectors

The set of all possible linear combinations of a given set of vectors.

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Basis of a Vector Space

A set of linearly independent vectors that span the entire vector space.

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Linear Transformation Definition

A function that maps vectors from one vector space to another, while maintaining the operations of addition and scalar multiplication.

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Dimension of a Vector Space

The number of vectors in a basis for the vector space, or the number of independent vectors needed to span the vector space.

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Linear Independence of Vectors

Vectors are linearly independent if no linear combination of one vector equals another member of the set, zero is only possible by equating all constants to 0.

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Rank-Nullity Theorem

For a linear transformation T: V → W, the sum of the dimension of the null space (nullity) and the dimension of the range space (rank) equals the dimension of the domain space (V).

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Vector Subspace

A subset of a vector space that is also a vector space under the same operations, meaning it's closed under addition and scalar multiplication.

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Zero Vector Space

Yes, the set containing only the zero vector forms a vector space, as it satisfies all the vector space axioms.

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Matrix: Vector Space?

Yes, the set of all m x n matrices over a field forms a vector space under matrix addition and scalar multiplication.

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R and R^n

R represents the set of all real numbers. R^n represents the set of all n-tuples of real numbers, forming a vector space with dimension n.

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Properties of Vector Space

Vector spaces have ten properties: closure under addition and scalar multiplication, associativity, commutativity, identity, inverse, distributivity.

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Union of Subspaces: Subspace?

The union of two subspaces is only a subspace if one subspace is contained within the other.

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What is a matrix?

A rectangular array of numbers, symbols, or expressions arranged in rows and columns.

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Determinant of a Matrix

A scalar value calculated from the elements of a square matrix. It represents the volume scaling factor of a linear transformation represented by the matrix.

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Rank of a matrix

The maximum number of linearly independent rows or columns in a matrix. It indicates the matrix's dimensionality.

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Cayley-Hamilton Theorem

States that every square matrix satisfies its own characteristic equation. This means that the matrix can be expressed as a polynomial in itself.

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Eigenvalue

A scalar that represents a scaling factor applied to an eigenvector when a linear transformation is applied.

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Eigenvector

A non-zero vector that doesn't change direction when a linear transformation is applied. Only its scale changes, determined by the eigenvalue.

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Characteristic polynomial

A polynomial obtained from a matrix, and its roots are the eigenvalues of that matrix.

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Green's Theorem

Relates a line integral around a closed curve to a double integral over the region enclosed by the curve. It connects line integrals to surface integrals.

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Study Notes

Course Information

  • Course Name: Calculus & Linear Algebra
  • Course Code: 21BTBS101
  • Faculty: MOHD SAGID ALI
  • Semester: FIRST
  • Session: 2023-2024
  • Question Bank (Unit 1)

Unit 1 (Calculus) - 2 Marks Questions

  • State Rolle's, Lagrange's Mean Value theorem
  • Write Maclaurin's series
  • Is the Rolle's theorem applicable to function tanx in (0,π/2)?
  • Can Rolle's theorem be applied to secx for [0, π]?
  • Write Euler's theorem
  • Write the geometrical meaning of Lagrange's Mean Value theorem
  • Write the geometrical meaning of Rolle's theorem
  • What is the meaning of partial derivative
  • Verify that (x^2/4 + y^4/4) / (x^1/5 + y^1/5) is a homogeneous function and find its degree
  • State Cauchy's mean value theorem
  • If x = acos³θ, y = bsin³θ, find (d²y/dx²)
  • How do you calculate maxima and minima of two variables x and y?
  • State all the derivatives involve in z = f(x, y)
  • If u = (x - y)² + (y − z)⁴ + (z – x)⁴, then find the value of (∂u/∂x) + (∂u/∂y) + (∂u/∂z)
  • Explain the graph given below which is related to Lagrange's mean value theorem

Unit 1 (Calculus) - 5 Marks Questions

  • verify langrange's theorem for the function f(x) = x(x-1)(x-2) in (0, 1/2) also find the value of c
  • Test x³ + y³ - 63(x + y) + 12xy for extrema points
  • Discuss the Maxima and Minima of the function 2(x² - y²) - x⁴ + y⁴
  • Expand the function f(x) =sinx about x=π/6 according to taylor series
  • verify Rolle's theorem and find the value of c. f(x) = x² - 8x + 5, [2,6]
  • Expand cosx by Taylor's theorem in powers of (x - π/4) upto fourth degree and find the coefficient of the last term.
  • If r² = x² + y², prove that (∂²r/∂x²) + (1/r) (∂r/∂x)²=(∂²r/∂y²) + (1/r)(∂r/∂y)²
  • Find (∂²z/∂x∂y) and (∂²z/∂y∂x) for the following function x² sin(y³) + xe³z - cos(z²) = 3y - 6z + 8
  • Expand √x by Taylor's theorem about point x=1 upto third degree. Hence find the value of √10
  • If z = x³ + y³ – 3axy, show that (∂²z/∂x∂y) = (∂²z/∂y∂x)
  • Let u = log(x² + y² + z²)¹/², then find (∂²u/∂x²)+(∂²u/∂y²)+(∂²u/∂z²)
  • Discuss the maxima and minima of x³ + y³ – 12x – 3y + 20
  • Find (df/dx) and (df/dy) for the following functions
    • f(x,y) = (xy – 1)²
    • f(x,y) = 1/(x+y)
    • f(x,y) = ln(x+y)
    • f(x,y) = sin²(x - 3y)
    • f(x,y) = xy
  • Find the maximum and minimum values of f(x, y) = x² + 3xy² - 15x² - 15y² + 72x
  • What is homogeneous function and state Euler's theorem in Partial derivatives
  • Write working rule to find the maximum and minimum values of two variable function f(x, y)
  • Find the first 3 terms in the Maclaurin series for (a) sin²x, (b) √(1−x²)

Unit 2 (Integral Calculus)

  • Find the area of ellipse (x²/a²) + (y²/b²) = 1
  • Find by double integration the area between y = x² and y = x
  • Evaluate ∫∫xy²dxdy (limits)
  • Evaluate ∫∫(x+y)dxdy (limits)
  • Evaluate ∫∫e^(x+y)dxdy (limits)
  • Evaluate ∫∫(x² + y²)dxdy (limits)
  • Evaluate ∫∫xyzdxdydz (limits)
  • Give physical interpretation of double integral
  • Give physical interpretation of triple integral
  • What is change of order of integration?
  • How do you calculate the area bounded by curves by double integration?
  • How do you calculate the volume of solids in double integral and in triple integral

Unit 3 (Vector Calculus)

  • State Gauss Theorem
  • State Green's Theorem
  • State Stoke's Theorem
  • Define scalar and vector point functions
  • Give examples of scalar and vector field
  • Define gradient of a scalar field
  • What do you mean by divergence of vector field
  • Give geometric interpretation of gradient
  • Write physical interpretation of divergence.
  • Define curl of vector F
  • What is the condition for irrotational vector?
  • When is a force said to be conservative.
  • If (x, y, z) = 3x²y-y³z², find ∇ø and |∇ ø| at (1, -2, -1)
  • Find a unit normal to the surface x²y + 2xz = 4 at the point (2, -2, 3)
  • Find the directional derivative of f = xyz² + 4xz² at (1, -2, -1) in the direction 2i – j-2k
  • Evaluate divergence of Ā = 2x²zi - xy²zj+3yz²k at the point (1, 1, 1)

Additional Information for Question Banks

  • Question Bank Units(2-3 questions are mentioned): Unit 02, Unit 03
  • Further multiple choice questions are also included in Question Banks, and section (A, B, and C)

Matrix and Determinants

  • Information on Matrices and Determinants, their applications, Rank, Cayley-Hamilton Theorem, Eigenvalues, Eigenvectors, Characteristic Polynomial of a Matrix.
  • Problems involving the mentioned concepts and finding the characteristic equation for a matrix.
  • Solving for the consistency and solution of systems of equations.

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