GRAM-SCHMIDT process converts a basis into
Athe eigenvectors of a matrix
Ban orthonormal basis
Cthe kernel of a matrix
Da diagonal matrix
Answer & Solution
Correct answer: B. an orthonormal basis
1. GRAM-SCHMIDT algorithm: takes a basis $\{v_1, v_2, \ldots, v_n\}$ and produces an ORTHONORMAL basis $\{u_1, u_2, \ldots, u_n\}$ that spans the same space.
2. STEP 1: $u_1 = v_1 / \|v_1\|$.
3. STEP $k$: subtract from $v_k$ the projections onto $u_1, \ldots, u_{k-1}$, then normalise.
4. The resulting vectors are MUTUALLY ORTHOGONAL and each have unit length.
5. Used to: build QR decomposition, find orthogonal projections, compute least squares solutions.
6. Other options confuse Gram-Schmidt with different operations.
_Source: Sergei Treil, "Linear Algebra Done Wrong", §5.2 (Gram-Schmidt)._
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