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MATH-C05-C-001

Add the equations to remove y.

Scale v_1 first, then add v_2.

A basis must span and avoid redundancy. Check whether [1, 1] can be made from the first two vectors.

Count independent output directions, not columns.

The column space is the horizontal line of vectors [c, 0].

Compute 2(-2) + 4(1).

Apply the matrix to [1, 0] and compare the result with [1, 0].

Matrix products act from right to left on a vector.

A rank-2 approximation keeps the two largest singular values.

PCA follows variance, not causality.

Use distributivity: A(x_0 + z) = Ax_0 + Az.

Compare the input dimension with the rank. Rank counts independent directions that remain visible in the output.

Singular values are strengths. A zero strength means that direction is collapsed.

Weak directions are not automatically useless. They may contain rare or task-specific detail.