Linear Algebra 3D

Question 1

What does a matrix transformation represent?

Accepted Answer

A function mapping vectors to new vectors — A matrix transformation is a function that maps input vectors to output vectors, performing operations like rotation, scaling, or shearing.

Question 2

What are eigenvalues?

Accepted Answer

Scalars showing the stretch factor — Eigenvalues are scalars that indicate the factor by which an eigenvector is stretched or compressed during a linear transformation.

Question 3

In machine learning, a single data example with n features is most naturally represented as…

Accepted Answer

A single n-dimensional feature vector — Each example is a point in n-dimensional space — a vector whose components are its feature values. A dataset stacks these as rows of a matrix.

Question 4

The dot product of two vectors a·b equals zero when the vectors are…

Accepted Answer

Orthogonal — a·b = |a||b|cos θ; it is zero exactly when cos θ = 0, i.e. the vectors are orthogonal. This underlies projections and decorrelation.

Question 5

For matrix multiplication A·B to be defined, the dimensions must satisfy…

Accepted Answer

columns of A = rows of B — An (m×n) matrix times an (n×p) matrix gives (m×p); the inner dimensions (n) must match. This is how layers transform feature vectors.

Question 6

What does the L2 (Euclidean) norm of a vector measure?

Accepted Answer

Its length/magnitude — The L2 norm is the straight-line length √(x₁²+…+xₙ²). L2 regularization penalizes large weight norms; distances in KNN use it.

Question 7

The L1 norm of a vector is…

Accepted Answer

The sum of absolute values — L1 = Σ|xᵢ|. L1 regularization (Lasso) tends to drive weights exactly to zero, producing sparse models — unlike L2.

Question 8

Cosine similarity between two vectors is computed as…

Accepted Answer

The dot product divided by the magnitudes — Cosine similarity is the dot product normalized by magnitudes, giving cos θ ∈ [−1,1]. It measures orientation, not length — central to embeddings/retrieval.

Question 9

The identity matrix I has the property that…

Accepted Answer

I·A = A·I = A — I leaves any conformable matrix unchanged under multiplication — the matrix analogue of the number 1.

Question 10

A matrix A is invertible only if its determinant is…

Accepted Answer

Non-zero — det(A) = 0 means A is singular (columns linearly dependent) and has no inverse. A non-zero determinant guarantees an inverse exists.

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