➕ Topic Summary
The Gram-Schmidt process is a method for orthonormalizing a set of vectors in an inner product space. Given a set of linearly independent vectors, it constructs an orthonormal basis for the span of those vectors. The process involves projecting each vector onto the subspace spanned by the previous vectors and subtracting that projection to obtain an orthogonal set. Then, each vector is normalized to obtain an orthonormal set.
Essentially, we start with a basis, make the vectors perpendicular (orthogonal), and then make them unit length (normalized). This process is fundamental in various areas of mathematics, physics, and engineering.
🔑 Part A: Vocabulary
Match the following terms with their definitions:
| Term |
Definition |
| 1. Orthonormal |
A. A vector with a length of 1. |
| 2. Linear Independence |
B. Vectors that are both orthogonal and normalized. |
| 3. Projection |
C. The component of one vector that lies along the direction of another. |
| 4. Unit Vector |
D. Vectors that cannot be written as a linear combination of each other. |
| 5. Orthogonal |
E. Vectors that are perpendicular to each other. |
(Answers: 1-B, 2-D, 3-C, 4-A, 5-E)
✍️ Part B: Fill in the Blanks
Complete the following paragraph using the words: basis, orthogonal, normalized, Gram-Schmidt, linearly independent.
The ______ process takes a set of ______ vectors and produces an ______ ______ set of vectors that span the same subspace. This process ensures that the resulting vectors form a ______ for the subspace, where each vector is ______ to all others and is ______.
(Answers: Gram-Schmidt, linearly independent, orthogonal, normalized, basis, orthogonal, normalized)
🤔 Part C: Critical Thinking
Explain why the Gram-Schmidt process requires the initial set of vectors to be linearly independent. What happens if they are not?