Linear Algebra - 1

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Linear Algebra

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AERO GATE TOPPER

Question Type: Multiple Choice Question Marks for correct answer: 1 | Negative Marks: 0.33

1. Which of the following is a requirement for a set of vectors to form a vector space?

Vectors must have integer components

The set must be closed under vector addition and scalar multiplication

All vectors must be orthogonal

The set must contain only non-zero vectors

Question Type: Multiple Choice Question Marks for correct answer: 1 | Negative Marks: 0.33

2. What does the null space of a matrix A represent?

The set of all vectors x such that A * x = 0

The set of all vectors x such that A * x = x

The set of all eigenvectors of A

The set of all orthogonal vectors to A

Question Type: Multiple Select Question Marks for correct answer: 2 | Negative Marks: 0

3. Which of the following statements about eigenvalues and eigenvectors are true?

Eigenvalues are scalars that scale eigenvectors

Eigenvectors corresponding to the same eigenvalue are always orthogonal

Eigenvectors corresponding to different eigenvalues are linearly independent

The sum of two eigenvectors is always an eigenvector

Question Type: Multiple Choice Question Marks for correct answer: 1 | Negative Marks: 0.33

4. What equation must be satisfied for λ to be an eigenvalue of a matrix A?

A * v = v

A * λ = v

det(A - λI) = 0

det(A - λA) = 1

Question Type: Multiple Choice Question Marks for correct answer: 1 | Negative Marks: 0.33

5. For an orthogonal matrix A, which of the following is true?

Aᵀ * A = A

Aᵀ * A = I

A * Aᵀ = 0

A⁻¹ * A = A

Question Type: Multiple Choice Question Marks for correct answer: 2 | Negative Marks: 0.67

6. The Gram-Schmidt process is used to:

Solve systems of linear equations

Calculate determinants of matrices

Orthogonalize a set of vectors

Diagonalize matrices

Question Type: Multiple Select Question Marks for correct answer: 2 | Negative Marks: 0

7. Which of the following statements about the rank-nullity theorem are true?

Rank(A) + Nullity(A) = Number of columns of A

Rank(A) is the number of linearly independent rows or columns

Nullity(A) is always greater than Rank(A)

Rank(A) is equal to the dimension of the null space of A

Question Type: Multiple Choice Question Marks for correct answer: 2 | Negative Marks: 0.67

8. A matrix A is diagonalizable if:

All of its eigenvalues are zero

It has a full set of linearly independent eigenvectors

It is a symmetric matrix

It is invertible

Question Type: Multiple Choice Question Marks for correct answer: 2 | Negative Marks: 0.67

9. Under what conditions does a system of linear equations have a unique solution?

The determinant of the coefficient matrix is non-zero

The number of equations is greater than the number of unknowns

The matrix is symmetric

The matrix has no eigenvalues

Question Type: Multiple Select Question Marks for correct answer: 2 | Negative Marks: 0

10. Which of the following statements about orthogonal matrices are true?

Aᵀ * A = I

Orthogonal matrices are always diagonalizable

The determinant of an orthogonal matrix is zero

Orthogonal matrices preserve vector lengths under transformations

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