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Let A be an 2x3 matrix. Then for the set {X ER2: Ax = 0), which one of the following is true? a) Subspace of R2 b) Subspace of R3 c) Both subspace of R3 and subspace of R2 d) Neither subspace of R3 nor subspace of R²
4. Let T be the linear operator on F which is represented in the standard ordered basis by the matrix c0 0 01 Let W be the nll space of T - c/. (a) Prove that W is the subspace spanned by 4 (b) Find the monic generators of the ideals S(u;W), S(q;W), s(G;W), 1 4. Let T be the linear operator on F which is represented in the standard ordered basis by the matrix c0 0 01 Let W...
Let A be an 4 x 3 matrix. Then for the set {x ∈ R3: Ax = 0), which one of the following is true? a) Subspace of R3 b) Subspace of R4 c) Both subspace of R3 and subspace of R4 d) Neither subspace of R3 nor subspace of R4
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector space. (b) Prove that L is an affine subspace of R3 (c) Compute a vector v such that L = v + W (6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y...
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector space. (b) Prove that L is an affine subspace of R3 (c) Compute a vector v such that L = v + W
Let I, Y ER" be two nonzero n-dimensional vectors and define the n x n matrix A = ty eigenvalues of A are 0 and y's Show that the
hint: H3. Let W1 = {ax? + bx² + 25x + a : a, b e R}. (a) Prove that W is a subspace of P3(R). (b) Find a basis for W. (c) Find all pairs (a,b) of real numbers for which the subspace W2 = Span {x} + ax + 1, 3x + 1, x + x} satisfies dim(W. + W2) = 3 and dim(Win W2) = 1. H3. (a) Use Theorem 1.8.1. (b) Let p(x) = ax +...
Example: Let x, y ∈ Rn, where n ∈ N. The line segment joining x to y is the subset {(1 − t)x + ty : 0 ≤ t ≤ 1 } of R n . A subset A of Rn, where n ∈ N, is called convex if it contains the line segment joining any two of its points. It is easy to check that any convex set is path-connected. (a) Let f : X → Y be an...
Let A be an m × n matrix The image of A is the set of vectors m(A) = {y : y = Ax for some x E Rn). which is a vector space The dimension of im(A) is called the rank of A, denoted by rank(A) (a) Find the rank of the matrix -62 1110 142 441 100-234 -1786478 46 -115 -46 -46 69 -122 85 150 174 -685 and enter in the box below rank(A) in应答 评分: 01...
Q5 Eigenmatrix 8 Points Let C12 M2 = 211 221 : Xij ER ER} 2 22 be the vector space of 2 x 2 real matrices with entrywise addition and scalar multiplication. Consider the subspace W = {X E M2 : X = XT} of M2 consisting of symmetric matrices. (a) (2pts) Find a basis of W. What is its dimension? 1 (b) (2pts) Let A= Show that if X EW then AXAT EW. (c) (4pts) Consider the linear transformation...