Q/ prove that:- If Vis a finite dimensional vector space, then this equivalence relation has only a single equivalence class.
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- Let V be the set of all positive real numbers. Determine whether V is a vector space with the operations shown below. x+y=xyAddition cx=xcScalar multiplication If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}28. Is C a real vector space? Explain.
- write in full sen6. P2 denotes the set of all polynomials of degree ≤ 2, in a single variable x, with real number coefficients. P2 is a vector space under the usual operations of polynomial addition (combining like terms) and scalar multiplication. Determine whether each set of polynomials in P2 is linearly independent or linearly dependent. If the set is linearly dependent, find a dependence relation. If the set is linearly independent, demonstrate that any dependence relation has to be trivial. (a) {2x, x+3} (b) {x, 2x+5, 10} (c) {x² + 5x+1, x²-7x, x² +10, 2x² + 3x-4} 1ST (d) Make a linearly independent set of three polynomials in P2. Explain how you know.Let P2 be the vector space of all polynomials of degree ≤ 2 with coefficients in R, andS= {1 + 2x, 1 + 3x + 5x^2 , 4x + 5x^2 } .Is Span(S)= P2?
- Determine which sets are vector spacesunder the given operations. For those that are not, list all axioms that failto hold. The set of all positive of real numbers x with the operations on x+ ´x =xx´ and kx = x^kI want the state of all vector space axioms that failan Given that { 91, 92, 93, 94} is a linearly independent set of vectors in av Vector space space V'( note that V need not be IR"). Use the diff Prove that {a an ax; -91 +92, 91-92 +93 ;-9, +92-93 +94"} is also a independence) linearly independent set (use the definition of Linearly
- Show that the set of all real polynomials with a degree n < 3 associated with the addition of polynomials and the multiplication of polynomials by a scalar form a vector space.Is the set of rational numbers with the usual definitions and scalar multiplication a vector space? Please show justification for answer and explain steps.4. Let W be a subspace of a vector space V. We define a relation: V₁ ~ V₂ if V₁ V₂ € W. V1 4a. Show that this relation '~' is an equivalence relation on V. Namely, show: (i) Reflexive: v~ v; (ii) Symmetric: v₁ ~ V2 ⇒ V2 ~ V₁; and (iii) Transitive: V₁ V2, V₂ V3 ⇒ V₁ ~ V3. 4b. Denote by V := = {x EV|x~v} the equivalence class contain- ing v (surely, v Ev), and we call v a representative of the class v. Show that v = v+W := {v + w|w€ W}. 4c. For v₂ EV, show: either V₁ V₂ = Ø or V₁ = V₂. Show: there are equivalence classes Va (a E A) such that V is a disjoint union of them (for some index set A): V = LlaEA Vaj show: V₁ = V₂ ⇒ V₁ ~ V₂. (Remark. 4c hold for any equivalence relation on a set.) 4d. Show that the cardinality Va| = |VB| for any a, ß € A. Hint. Use 4b and show the bijectivity of the map: f: Va → VB (Va+W+Vg+w).