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Mathematics & Statistics

Vector and Matrices

Interview questions on Vector and Matrices.

51 questions

All subtopicsVectorMatrices

Vector

Q1. Define following terms

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  1. Scalers
  2. Vectors
  3. Matrices
  4. Tensors

Vector

Q2. What is broadcasting in matrices in context of deep learning?

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Vector

Q3. Dot product 1. What’s the geometric interpretation of the dot product of two vectors? 1. Given a vector $u $, find vector$ v $of unit length such that the dot product of$ u $and$ v$ is maximum.

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Vector

Q4. Outer product

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  1. Given two vectors a=[3,2,1]a=[3,2,1] and b=[−1,0,1]b=[−1,0,1]. Calculate the outer productaTb a^Tb ?
  2. Give an example of how the outer product can be useful in ML.

Vector

Q5. What does it mean for two vectors to be linearly independent?

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Vector

Q6. Given two sets of vectors $A=a_1,a_2,a_3,...,a_n$ and $B=b_1,b_2,b_3,...,b_m$. How do you check that they share the same basis?

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Vector

Q9. Given $n$ vectors, each of $d$ dimensions. What is the dimension of their span?

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Vector

Q10. Norms and metrics

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  1. What's the norm? What is L0,L1,L2,LnormL_0,L_1,L_2,L_{norm}?
  2. How do norms and metrics differ? Given a norm, make a metric. Given a metric, can we make a norm?

Matrices

Q12. Define the condition under which we can multiply two matrices?

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Matrices

Q15. How is the Hadamard product different from the dot product?

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Matrices

Q17. Is dot product between two vectors are commutative?

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Matrices

Q25. Why do we say that matrices are linear transformations?

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Matrices

Q26. Do all matrices have an inverse? Is the inverse of a matrix always unique?

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Matrices

Q29. When we should use $L^1$ norm instead of $L^2$ norm?

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Matrices

Q32. Can you write the dot product of two vectors in terms of their norms?

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Matrices

Q33. Define following type of matrices:

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  • Diagonal Matrix
  • Symmetric Matrix
  • Orthogonal Matrix

Matrices

Q36. Express the eigen decomposition of square matrix $\mathbf{A}$ ?

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Matrices

Q39. What is the benefit of using Singular Value Decomposition(SVD) over eigenvalue decomposition?

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Matrices

Q41. What is the relationship between eigen-value decomposition and SVD?

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Matrices

Q45. What does the absolute value of the determinant depicts?

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Matrices

Q46. What happens to the determinant of a matrix if we multiply one of its rows by a scalar $t×R$ ?

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Matrices

Q47. A $4×4$ matrix has four eigenvalues $3, 3, 2, −1$. What can we say about the trace and the determinant of this matrix?

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Matrices

Q48. Given the following matrix:

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[14−2−13235−6]\begin{bmatrix} 1 & 4 & -2 \\\\ -1 & 3 & 2 \\\\ 3 & 5 & -6 \end{bmatrix} Without explicitly using the equation for calculating determinants, what can we say about this matrix’s determinant?

Matrices

Q49. What’s the difference between the covariance matrix $A^TA$ and the Gram matrix $AA^T$?

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Matrices

Q50. Given $A∈R^{n×m}$ and $b∈R^n$1. Find$ x $such that:$ Ax=b$.

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  1. When does this have a unique solution?
  2. Why is it when AA has more columns than rows, Ax=bAx=b has multiple solutions?
  3. Given a matrix AA with no inverse. How would you solve the equation Ax=bAx=b? What is the pseudo inverse and how to calculate it?

Matrices

Q51. Given a very large symmetric matrix $A$ that doesn’t fit in memory, say $A∈R^{1M×1M}$ and a function $f$ that can quickly compute $f(x)=Ax$ for $x∈R1M$. Find the unit vector$ x $so that$ x^TAx$ is minimal.

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