6 Questions
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 |.
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).
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.
(a) The rank of a matrix is the maximum number of linearly independent rows or columns. (b) The echelon form of a matrix is a specific row-echelon form obtained through row operations. (c) The normal form of a matrix refers to the row-echelon form obtained through elementary row operations.
Write the canonical form of the matrix: | 2 2 2 \ 4 3 0 \ 3 1 -4 |.
The canonical form is: | 1 0 0 \ 0 1 0 \ 0 0 0 |.
Prove that the transformation T(x₁, x₂, x₃) = (x₁ - x₂ + 2x₃, 2x₁ + x₂, -x₁ - 2x₂ + 2x₃) is a linear transformation.
The transformation T satisfies the properties of additivity and homogeneity, hence it is a linear transformation.
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.
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.
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