Subject GRE-Style: Orthogonal Projection

GRE Math Subject · Axiom Academy

EXAMPLE Subject GRE-Style: Orthogonal Projection Computing the projection of a vector onto a subspace Let and let be the subspace of spanned by Question: What is the orthogonal projection of onto ? The projection of onto (with basis ) is: First, compute the inner products (using dot product in ): Verify: This result is a scalar multiple of (so it lies in ). ✓ Also, should be orthogonal to : Conceptual understanding: The projection "squashes" the vector onto the line spanned by . The formula weights the basis vector by the cosine of the angle between and . Key lesson: Always check orthogonality: for every basis vector.

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