Math 55 - Unit 1: Linear Algebra & Real Analysis
Problem 1: Advanced Linear Algebra Concept
This problem explores a foundational concept in Unit 1 of Math 55, such as vector spaces, matrix
transformations, or eigenvalues. For example:
Let A be a 3x3 matrix with real entries. Prove that if A is orthogonal, then its determinant is +/-1.
Answer:
Orthogonal matrices preserve length and angle, meaning A-transpose * A = Identity. Taking determinants on
both sides:
det(A-transpose * A) = det(I) => det(A-transpose) * det(A) = 1 => (det A)^2 = 1 => det A = +/-1.
This result is crucial in understanding the behavior of orthogonal transformations in R^n.
Problem 2: Advanced Linear Algebra Concept
This problem explores a foundational concept in Unit 1 of Math 55, such as vector spaces, matrix
transformations, or eigenvalues. For example:
Let A be a 3x3 matrix with real entries. Prove that if A is orthogonal, then its determinant is +/-1.
Answer:
Orthogonal matrices preserve length and angle, meaning A-transpose * A = Identity. Taking determinants on
both sides:
det(A-transpose * A) = det(I) => det(A-transpose) * det(A) = 1 => (det A)^2 = 1 => det A = +/-1.
This result is crucial in understanding the behavior of orthogonal transformations in R^n.
Page 1
, Math 55 - Unit 1: Linear Algebra & Real Analysis
Problem 3: Advanced Linear Algebra Concept
This problem explores a foundational concept in Unit 1 of Math 55, such as vector spaces, matrix
transformations, or eigenvalues. For example:
Let A be a 3x3 matrix with real entries. Prove that if A is orthogonal, then its determinant is +/-1.
Answer:
Orthogonal matrices preserve length and angle, meaning A-transpose * A = Identity. Taking determinants on
both sides:
det(A-transpose * A) = det(I) => det(A-transpose) * det(A) = 1 => (det A)^2 = 1 => det A = +/-1.
This result is crucial in understanding the behavior of orthogonal transformations in R^n.
Problem 4: Advanced Linear Algebra Concept
This problem explores a foundational concept in Unit 1 of Math 55, such as vector spaces, matrix
transformations, or eigenvalues. For example:
Let A be a 3x3 matrix with real entries. Prove that if A is orthogonal, then its determinant is +/-1.
Answer:
Orthogonal matrices preserve length and angle, meaning A-transpose * A = Identity. Taking determinants on
both sides:
det(A-transpose * A) = det(I) => det(A-transpose) * det(A) = 1 => (det A)^2 = 1 => det A = +/-1.
This result is crucial in understanding the behavior of orthogonal transformations in R^n.
Page 2
, Math 55 - Unit 1: Linear Algebra & Real Analysis
Problem 5: Advanced Linear Algebra Concept
This problem explores a foundational concept in Unit 1 of Math 55, such as vector spaces, matrix
transformations, or eigenvalues. For example:
Let A be a 3x3 matrix with real entries. Prove that if A is orthogonal, then its determinant is +/-1.
Answer:
Orthogonal matrices preserve length and angle, meaning A-transpose * A = Identity. Taking determinants on
both sides:
det(A-transpose * A) = det(I) => det(A-transpose) * det(A) = 1 => (det A)^2 = 1 => det A = +/-1.
This result is crucial in understanding the behavior of orthogonal transformations in R^n.
Problem 6: Advanced Linear Algebra Concept
This problem explores a foundational concept in Unit 1 of Math 55, such as vector spaces, matrix
transformations, or eigenvalues. For example:
Let A be a 3x3 matrix with real entries. Prove that if A is orthogonal, then its determinant is +/-1.
Answer:
Orthogonal matrices preserve length and angle, meaning A-transpose * A = Identity. Taking determinants on
both sides:
det(A-transpose * A) = det(I) => det(A-transpose) * det(A) = 1 => (det A)^2 = 1 => det A = +/-1.
This result is crucial in understanding the behavior of orthogonal transformations in R^n.
Page 3
Problem 1: Advanced Linear Algebra Concept
This problem explores a foundational concept in Unit 1 of Math 55, such as vector spaces, matrix
transformations, or eigenvalues. For example:
Let A be a 3x3 matrix with real entries. Prove that if A is orthogonal, then its determinant is +/-1.
Answer:
Orthogonal matrices preserve length and angle, meaning A-transpose * A = Identity. Taking determinants on
both sides:
det(A-transpose * A) = det(I) => det(A-transpose) * det(A) = 1 => (det A)^2 = 1 => det A = +/-1.
This result is crucial in understanding the behavior of orthogonal transformations in R^n.
Problem 2: Advanced Linear Algebra Concept
This problem explores a foundational concept in Unit 1 of Math 55, such as vector spaces, matrix
transformations, or eigenvalues. For example:
Let A be a 3x3 matrix with real entries. Prove that if A is orthogonal, then its determinant is +/-1.
Answer:
Orthogonal matrices preserve length and angle, meaning A-transpose * A = Identity. Taking determinants on
both sides:
det(A-transpose * A) = det(I) => det(A-transpose) * det(A) = 1 => (det A)^2 = 1 => det A = +/-1.
This result is crucial in understanding the behavior of orthogonal transformations in R^n.
Page 1
, Math 55 - Unit 1: Linear Algebra & Real Analysis
Problem 3: Advanced Linear Algebra Concept
This problem explores a foundational concept in Unit 1 of Math 55, such as vector spaces, matrix
transformations, or eigenvalues. For example:
Let A be a 3x3 matrix with real entries. Prove that if A is orthogonal, then its determinant is +/-1.
Answer:
Orthogonal matrices preserve length and angle, meaning A-transpose * A = Identity. Taking determinants on
both sides:
det(A-transpose * A) = det(I) => det(A-transpose) * det(A) = 1 => (det A)^2 = 1 => det A = +/-1.
This result is crucial in understanding the behavior of orthogonal transformations in R^n.
Problem 4: Advanced Linear Algebra Concept
This problem explores a foundational concept in Unit 1 of Math 55, such as vector spaces, matrix
transformations, or eigenvalues. For example:
Let A be a 3x3 matrix with real entries. Prove that if A is orthogonal, then its determinant is +/-1.
Answer:
Orthogonal matrices preserve length and angle, meaning A-transpose * A = Identity. Taking determinants on
both sides:
det(A-transpose * A) = det(I) => det(A-transpose) * det(A) = 1 => (det A)^2 = 1 => det A = +/-1.
This result is crucial in understanding the behavior of orthogonal transformations in R^n.
Page 2
, Math 55 - Unit 1: Linear Algebra & Real Analysis
Problem 5: Advanced Linear Algebra Concept
This problem explores a foundational concept in Unit 1 of Math 55, such as vector spaces, matrix
transformations, or eigenvalues. For example:
Let A be a 3x3 matrix with real entries. Prove that if A is orthogonal, then its determinant is +/-1.
Answer:
Orthogonal matrices preserve length and angle, meaning A-transpose * A = Identity. Taking determinants on
both sides:
det(A-transpose * A) = det(I) => det(A-transpose) * det(A) = 1 => (det A)^2 = 1 => det A = +/-1.
This result is crucial in understanding the behavior of orthogonal transformations in R^n.
Problem 6: Advanced Linear Algebra Concept
This problem explores a foundational concept in Unit 1 of Math 55, such as vector spaces, matrix
transformations, or eigenvalues. For example:
Let A be a 3x3 matrix with real entries. Prove that if A is orthogonal, then its determinant is +/-1.
Answer:
Orthogonal matrices preserve length and angle, meaning A-transpose * A = Identity. Taking determinants on
both sides:
det(A-transpose * A) = det(I) => det(A-transpose) * det(A) = 1 => (det A)^2 = 1 => det A = +/-1.
This result is crucial in understanding the behavior of orthogonal transformations in R^n.
Page 3
