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- (b) Show that the set of all positive real numbers, with x+y and cx redefined to equal the usual xy and x, is a vector space, or indicate why it is not. What is the "zero vector"?arrow_forward2. Determine whether each set described is a subspace or R2. Justify your answer. H (a) The set of all vectors of the form (b) The set of all vectors of the form where can be any real number. X [ ] 2x where can be any real number.arrow_forwardThe functions f(x) = x2 and g(x) = 5x are "vectors" in F. This is the vector space of all real functions. (The functions are defined for -oo < x < oo.) The combination 3f(x) - 4g(x) is the function h(x) = __ .arrow_forward
- Determine False R¹ is a subspace of R¹7 whether the statement is true or false.arrow_forwardWhich of the following are vector spaces? Justify your answer. (a) The set of all polynomials of degree 3. (b) The set of all vectors x = (x1, x2, x3), satisfying 3x₁ + 5x2 − 9x3 = 2023. (c) The set of all vectors x = (x1, x2, x3), satisfying 2024x₁ + x2 = 0 has a unique solution. (d) The set of all 3 × 3 matrices such that Ax= (e) The set of all n × n (n = N) diagonal matrices. - x3 = 0. -arrow_forward(3) For each of the following sets, determine whether it is linearly independent or dependent. (0) € 9000 3 (c) {x³x, 2x² + 4,-2x³ +3x²+2x+6} in P₂ (R), where P3 (R) denotes the vector space of polynomials over the reals of degree at most 3. -1 2 (d) {(-12 1) (1 7¹). (1¹3) (3¹2)} in M2x2(R), where M2x2(R) 0 denotes the vector space of 2 × matrices over the reals. {0·0)} span(S), i.e., orthogonal to every vector in span(S). (b) (4) Let S in R4. 7 in R4. CR4. Find all vectors in R4 that are orthogonal to (5) Let A and B be matrices over the reals. For each of the following statements, determine whether it is true or false. If it is true, prove it. If it is false, give a counter example to disprove it. (a) If A + B is defined, then rank(A + B) = rank(A) + rank(B). (b) If AB is defined, then rank(AB) = rank(A) rank(B). (c) If A has size m × n, then rank(A) ≤ min{m, n}. (d) rank(A) = rank(At) (e) nullity(A) = nullity(A¹)arrow_forward
- Let A = 2 -5 6 ✓ 5 -2 6 b = Select true or false for each statement. True -2 [3]. False and c = 1. The vector c is in the range of T 2. The vector b is in the kernel of T -3 [3³] Define T(x) = Ax.arrow_forwardThe Hamacher product is the t-norm given by ab (а,b) %3 а+b-ab Show that this function satisfiest-norm operator properties.arrow_forwardProblem 8 Let V be the set of differentiable real-valued functions with domain R. Prove that V is a subspace of the set of functions F(R, R). (You may quote anything you like from elementary calculus without proof)arrow_forward
- Are all functions from one vector space to another linear?arrow_forwardI need help for problem (h). Check that the set at (h) is a subspace of Rn or not.arrow_forwardLet B={(1,-2,0,-1), (0,1,0,3), (1,2,1.0)], and x=(4,3,-2,-5) Find [x]B Give your answer in the form (a,b,c) with no spacesarrow_forward
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