Podcast
Questions and Answers
Prove the condition for a non-empty subset to be a subspace of a vector space.
Prove the condition for a non-empty subset to be a subspace of a vector space.
The subset W of a vector space V (F) is a subspace of V if and only if ax + by is in W for all real numbers a, b and all vectors x, y in W.
Find the eigenvalues and eigenvectors of the matrix A = | 2 3 \ 1 4 |.
Find the eigenvalues and eigenvectors of the matrix A = | 2 3 \ 1 4 |.
The eigenvalues are 1 and 5, with corresponding eigenvectors [1, -1] and [3, 1].
Apply the Gram-Schmidt process to obtain an orthonormal set of vectors from P₁ = (1,0,1), P₂ = (1,0,1), P₃ = (0,3,4).
Apply the Gram-Schmidt process to obtain an orthonormal set of vectors from P₁ = (1,0,1), P₂ = (1,0,1), P₃ = (0,3,4).
The orthonormal vectors are P₁' = (0.577, 0, 0.577), P₂' = (0.577, 0, -0.577), P₃' = (-0.516, 0.774, 0.361).
Define the rank, echelon form, and normal form of a matrix.
Define the rank, echelon form, and normal form of a matrix.
Signup and view all the answers
Write the canonical form of the matrix: | 2 2 2 \ 4 3 0 \ 3 1 -4 |.
Write the canonical form of the matrix: | 2 2 2 \ 4 3 0 \ 3 1 -4 |.
Signup and view all the answers
Prove that the transformation T(x₁, x₂, x₃) = (x₁ - x₂ + 2x₃, 2x₁ + x₂, -x₁ - 2x₂ + 2x₃) is a linear transformation.
Prove that the transformation T(x₁, x₂, x₃) = (x₁ - x₂ + 2x₃, 2x₁ + x₂, -x₁ - 2x₂ + 2x₃) is a linear transformation.
Signup and view all the answers
Study Notes
Vector Space and Subspace
- A non-empty subset W of a vector space V is a subspace of V if it satisfies the following condition: W is closed under vector addition and scalar multiplication.
- In other words, for all u, v in W and a in ℝ, u + v is in W and au is in W.
Eigenvalues and Eigenvectors
- The eigenvalues of the matrix A = | 2 3 \ 1 4 | are λ₁ = 5 and λ₂ = 1.
- The eigenvectors corresponding to λ₁ = 5 are vectors of the form (3t, t), where t is a non-zero real number.
- The eigenvectors corresponding to λ₂ = 1 are vectors of the form (t, -t), where t is a non-zero real number.
Gram-Schmidt Process
- Applying the Gram-Schmidt process to P₁ = (1,0,1), P₂ = (1,0,1), P₃ = (0,3,4) yields an orthonormal set of vectors:
- Q₁ = (1/√2, 0, 1/√2)
- Q₂ = (0, 0, 0) (since P₂ is a linear combination of P₁)
- Q₃ = (3/√10, 1/√10, 2/√10)
Matrix Operations
- The rank of a matrix is the maximum number of linearly independent rows or columns.
- The echelon form of a matrix is a row-equivalent matrix with leading entries of 1, and all other entries in the same column are 0.
- The normal form (or reduced row echelon form) of a matrix is an echelon form with all entries below the leading entries being 0.
Canonical Form
- The canonical form of the matrix | 2 2 2 \ 4 3 0 \ 3 1 -4 | is:
- | 1 0 0 \ 0 1 0 \ 0 0 1 |
Linear Transformation
- A transformation T: ℝ³ → ℝ³ is a linear transformation if it satisfies the following conditions:
- T(u + v) = T(u) + T(v) for all u, v in ℝ³
- T(au) = aT(u) for all u in ℝ³ and a in ℝ
- The transformation T(x₁, x₂, x₃) = (x₁ - x₂ + 2x₃, 2x₁ + x₂, -x₁ - 2x₂ + 2x₃) is a linear transformation.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Prepare for the end semester examination with these practice questions on vector spaces, subspaces, eigenvalues, and eigenvectors. Explore concepts such as proving a subset is a subspace and finding eigenvalues and eigenvectors of a matrix.
