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Questions and Answers
What makes two vectors orthogonal?
What makes two vectors orthogonal?
If their dot product is 0.
Which of the following statements is true regarding the Fundamental Theorem of Linear Algebra Pt. 2?
Which of the following statements is true regarding the Fundamental Theorem of Linear Algebra Pt. 2?
What is the orthogonal complement of subspace V?
What is the orthogonal complement of subspace V?
Contains every vector that is perpendicular to V.
What condition does an orthogonal matrix Q satisfy?
What condition does an orthogonal matrix Q satisfy?
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What happens when Q is square?
What happens when Q is square?
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What is the Gram-Schmidt process?
What is the Gram-Schmidt process?
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What is the setup for the Normal Equations?
What is the setup for the Normal Equations?
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How is error defined in the context of regression?
How is error defined in the context of regression?
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What is the first step of the Gram-Schmidt process?
What is the first step of the Gram-Schmidt process?
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What is the second step of the Gram-Schmidt process?
What is the second step of the Gram-Schmidt process?
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What is the A = LU (LDU) factorization?
What is the A = LU (LDU) factorization?
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How is the dimension defined in linear algebra?
How is the dimension defined in linear algebra?
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Study Notes
Orthogonality of Vectors
- Two vectors are orthogonal if their dot product equals zero (v^Tw = 0).
- The relationship ||v||² + ||w||² = ||v + w||² holds for orthogonal vectors.
Fundamental Theorem of Linear Algebra Pt. 2
- The row space (row(A)) and null space (null(A)) are orthogonal complements.
- The column space (col(A)) and the null space of the transpose (null(A^T)) are also orthogonal complements.
Orthogonal Complement of a Subspace
- Contains all vectors that are perpendicular to the subspace V, denoted as V perp.
Orthogonal Matrix Properties
- An orthogonal matrix Q satisfies the equation (Q^T)Q = I, where Q^T is the transpose of Q.
Square Orthogonal Matrix
- For a square orthogonal matrix, (Q^T)Q = I implies that Q^T = Q⁻¹ (the transpose is equal to the inverse).
Gram-Schmidt Process
- The Gram-Schmidt process transforms n independent vectors into a corresponding set of orthogonal vectors.
- The output is represented as A = QR, where Q contains orthogonal vectors.
Normal Equations Setup
- The normal equations are represented as: (t,b) ==> C + Dt = b.
- Expressed in matrix form, the normal equations are (A^T)Ax° = (A^T)b.
Error in Linear Equations
- The prediction p is determined by p = Ax.
- The error between the actual output b and prediction p is expressed as e = b - p.
First Step of Gram-Schmidt
- The first step computes B as follows: B = b - [(A^T)b/(A^T)A]A, ensuring B is orthogonal to A.
Second Step of Gram-Schmidt
- The second step computes C as C = c - [(A^T)c/(A^T)A]A - [(B^T)c/(B^T)B]B, making C orthogonal to both A and B.
A = LU (LDU) Factorization
- In LU factorization:
- L is a lower triangular matrix with 1's on the diagonal.
- U is an upper triangular matrix, with pivots on the diagonal.
- D consists of only the pivots.
Dimension in Linear Algebra
- The dimension is calculated by counting the number of independent columns in the matrix.
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Description
This quiz covers key concepts related to orthogonality in linear algebra. Topics include the definition of orthogonal vectors, properties of orthogonal matrices, the Gram-Schmidt process, and the relationships between subspaces. Test your understanding of these fundamental ideas and their implications in linear algebra.
