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- The 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_forwardDo questions 53 and 54 Show if it is a subspace using these 3 steps: 1. has to be equal to the 0 vector 2. has to be closed under addition 3. has to be closed under mulitplicationarrow_forwardSolve the following questions: Q1) Check whether the set W = {(2a + 5b, –a + b, 3a + 4b); a, b are real numbers } is a subspace of R3 or not.arrow_forward
- Determine False R¹ is a subspace of R¹7 whether the statement is true or false.arrow_forwardI need help for problem (h). Check that the set at (h) is a subspace of Rn or not.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_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_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_forwardSuppose y1 ( x), y2 ( x), y3 ( x) are three different functions of x. The vector space they span could have dimension 1, 2, or 3. Give an example of y1, y2, y3 to show each possibility.arrow_forward
- each statement is true or false. You do not need to give a reason. Problem 1. Consider the vector space P2 of real polynomials of degree at most 2. The set W = {p(x) | p(0) = 1} is a subspace of P2.arrow_forwardWhich of the following constraints results in the set of all polynomials of the form ao + a₁ + a₂x² forms a subspace of P₂ (R)? Select all the correct options. ao = a₁ = a2 = 0 a₁ = 0 ao ≥ 0 2a1a2arrow_forwardLet 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_forward
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