Math 270 Test 3 Review: Subspaces H And K Of A Vector

Math 270 Test 3 Review1 Given Subspaces H And K Of A Vector Space V

Given subspaces H and K of a vector space V, the sum of H and K, written as H + K, is the set of all vectors in V that can be written as the sum of two vectors, one in H and the other in K; that is H + K = {w : w = u + v, where u ∈ H and v ∈ K}. Show that H + K is a subspace of V.

Show that H is a subspace of H + K and K is a subspace of H + K.

Find an explicit description of Nul A by listing vectors that span the null space for A = [1 - 3; 4 0].

Let A = [-6 12 -3 6] and w = [2 1]. Determine if w ∈ Col A and if w ∈ Nul A.

Define T : P2 → R2 by T(p) = [p(0) p(1)]. Show that T is a linear transformation. For p(t) = 3 + 5t + 7t², find T(p).

Find a basis for the space spanned by vectors v₁ = [1, 0, -3, 2], v₂ = [-3], v₃ = [-3, -1], v₄ = [1, -3, -8, 7], v₅ = [2, 1, -6, 9].

Given vectors v₁ = [4, -3, 7], v₂ = [19, -2], v₃ = [−1, ?], and H = Span{v₁, v₂, v₃}, use the relation 4v₁ + 5v₂ − 3v₃ = 0 to find a basis for H.

Find the coordinate vector [x]_B of x = [8, -9, 6] relative to basis B = {b₁, b₂, b₃} with b₁ = [1, -1, -3], b₂ = [−1], b₃ = [2, -2, 4].

Use the inverse matrix to find [x]_B for x = [2, -6] with basis B = {[3, -5], [−4, 6]}.

Set B = {1 + t², t + t², 1 + 2t + t²} as a basis for P₂. Find the coordinate vector of p(t) = 1 + 4t + 7t² relative to B.

Using coordinate vectors, test the linear independence of the set of polynomials 1 + 2t³, 2 + t − 3t², −t + 2t² − t³. Explain your work.

Find the dimension of Nul A and Col A for the matrix A = [1 -6 9; 0 13 0].

Assuming matrices A and B are row equivalent, find bases for Col A, Row A, and Nul A for matrices A and B given below (not repeated here). Also, find bases for these subspaces.

If matrix A is 3×8 with rank 3, find dim(Nul A), dim(Row A), and rank(AT).

Let A = {a₁, a₂, a₃} and B = {b₁, b₂, b₃} be bases for a vector space V, with relations a₁ = 4b₁ − b₂, a₂ = −b₁ + b₂ + b₃, and a₃ = b₂ − 2b₃. Find the change-of-coordinate matrix from A to B and find [x]_B for x = 3a₁ + 4a₂ + a₃.

Given bases B = {b₁, b₂} and C = {c₁, c₂} for R², find the change-of-coordinate matrices from B to C and from C to B, with the basis vectors provided.

In P₂, find the change-of-coordinate matrix from basis B = {1−2t + t², 3−5t + 4t², 2t + 3t²} to the standard basis C = {1, t, t²}. Then find the coordinate vector of −1 + 2t relative to B.

Find a basis for the eigenspace corresponding to eigenvalue λ of matrix A = [[4, 2], [3, -1]]. Also, find the characteristic polynomial and eigenvalues of A.

Find the characteristic polynomial of matrix [[6, -2], [0, -1]], and the eigenvalues of the matrix.

Paper For Above instruction

In linear algebra, the concept of subspaces and their properties are fundamental to understanding the structure of vector spaces. The sum of two subspaces H and K within a vector space V is defined as the set of all vectors that can be expressed as the sum of one vector from each subspace. Mathematically, H + K = {w ∈ V : w = u + v, where u ∈ H and v ∈ K}. To demonstrate that H + K is itself a subspace, one must verify the closure under vector addition and scalar multiplication, as well as the presence of the zero vector. Since both H and K are subspaces, 0 vectors are contained within each, and hence their sum includes the zero vector. When adding two vectors from H + K, the sums remain within H + K because of the linearity of vector addition in the subspaces. Likewise, scalar multiples of vectors in H + K stay within H + K because both H and K are closed under scalar multiplication. Consequently, H + K satisfies all the criteria of a subspace, confirming its status as a subspace of V.

Furthermore, H and K are subspaces of H + K because they are subsets closed under addition and scalar multiplication, with their zero vectors included. Specifically, every element of H can be written as u + 0, with u ∈ H, demonstrating the inclusion H ⊆ H + K. A similar argument holds for K. These factoids underscore the hierarchical structure of subspaces within larger vector spaces, showcasing their interactions and the utility of the sum operation.

For example, consider the matrix A = [[1, -3], [4, 0]]. To determine the null space, Nul A, one finds all vectors x such that Ax = 0. Solving the homogeneous system yields a basis for Nul A: vectors that span the null space. For A, this solution typically involves setting free variables and expressing the dependent variables accordingly, leading to a parametrized form. The vectors derived from this parametrization form a basis for the null space, illustrating the vectors that nullify the transformation represented by A.

Considering the vector w = [2, 1], the question arises if it belongs to the column space of A, Col A, or to the null space, Nul A. Determining whether w ∈ Col A involves solving Ax = w for x, verifying the existence of a solution. If a solution exists, then w is in Col A. To check whether w ∈ Nul A, one must see if Aw = 0. If so, then w is in the null space of A. In this case, w simultaneously resides in both subspaces, indicating underlying linear relationships captured by the matrix.

Linear transformations from polynomial spaces to Euclidean spaces often utilize evaluation functions such as T(p) = [p(0), p(1)]. This linear operator maps each polynomial p in P₂ to the vector in R² given by its evaluations at 0 and 1. Validation of linearity involves demonstrating additivity and scalar compatibility—e.g., T(p + q) = T(p) + T(q) and T(cp) = cT(p)—which follow from properties of polynomial evaluation. Such mappings are instrumental in connecting polynomial spaces and finite-dimensional vector spaces, offering insights into their structure and transformations.

Finding a basis for a set of vectors involves determining a minimal set of linearly independent vectors that span the same space. For vectors v₁, v₂, v₃, v₄, v₅, one employs row reduction techniques on the matrix whose rows are the vectors to identify which vectors form a basis. Redundant vectors are eliminated, and the remaining independent vectors constitute the basis. This process confirms independence and minimal spanning sets, fundamental in vector space analysis.

In subspace analysis, linear dependence relations like 4v₁ + 5v₂ − 3v₃ = 0 reveal how vectors can be expressed as linear combinations of others, simplifying the basis. Given such relations, one can choose vectors from the original set that do not satisfy this relation or explicitly express one vector in terms of others to form a basis for H. This procedure ensures the basis spans the subspace while maintaining independence.

Expressing vectors in a coordinate system relative to a basis B involves solving systems of equations to find the coefficients in the basis expansion. For example, to find [x]_B for x in the span of B = {b₁, b₂, b₃}, set up and solve the system x = c₁b₁ + c₂b₂ + c₃b₃ for the coefficients c₁, c₂, c₃. When an inverse matrix exists, the process reduces to multiplying the inverse of the basis matrix by the vector x, streamlining the calculation of coordinates.

The change-of-coordinate matrices allow translation between different bases, facilitating coordinate transformations. Given bases B and C, the matrix from C to B is obtained by expressing each basis vector of C as linear combinations of the basis vectors of B. The inverse of this matrix provides the transformation from B to C. These matrices are critical for coordinate change in vector spaces.

In polynomial spaces, the basis transformation involves expressing polynomials with respect to one basis in terms of the standard basis. For B = {1−2t + t², 3−5t + 4t², 2t + 3t²}, finding the change-of-coordinate matrix entails expressing each basis vector in terms of the standard basis {1, t, t²}. Consequently, the coordinate vector of a polynomial such as −1 + 2t relative to B is obtained by solving the relevant systems or multiplying by the inverse matrix.

Eigenvalues and eigenspaces of matrices reveal the modes of the linear transformation, characterized by solutions to (A − λI)v = 0. For a given eigenvalue λ, the eigenspace is the null space of (A − λI). Computing the eigenvalues involves analyzing the characteristic polynomial det(A − λI), and the eigenvectors are obtained from the null space. These concepts are central to spectral theory and matrix decompositions.

The characteristic polynomial of a matrix is derived from the determinant of (A − λI). Its roots, the eigenvalues, provide insight into the matrix's structure. For example, a matrix like [[3, −2], [1, −1]] yields a quadratic polynomial in λ, with roots indicating the eigenvalues—here, computed via the characteristic polynomial—identifying key spectral properties of the matrix.

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