Duplicating the method of proof of Theorem 10,show that P(A∩B∩C∩D) = P(A)·P(B|A)·P(C|A∩B)·P(D|A ∩ B ∩ C) provided that P(A ∩ B ∩ C) Z 0.

Q: 1. Consider the set V of all polynomials of degree two or less, along with the following definitions…

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Q: 1 -1 0 2 Find a mat rix P such that P 1 0 0 2

A: consider P be any matrix having entries P=abcdgiven: P-1 1-102P=1002multypling P from right side.P…

Q: a) Find P(X1 P(X21 = 1|X19 = 2). = 2|Xo = 0), P(X4 1|X3 = 0, X2 = 1), P(X2 = 1|Xo = 1) and %3D %3D…

A: Given :P=0.70.20.10.30.50.20.10.40.5P2=0.560.280.160.380.390.230.240.420.34

Q: Exercise 195 Further problems on Leibniz's theorem 7. Given 2x2y" + xy' +3y 0 show that 2x2 y(n+2) +…

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Q: The domain of the functi f (x, y, z) = In(3.xy – 6) {(x, y, z) e R³ : xy 20 {(x, y) € R² : xy 2

A: Consider the function  f(x,y,z)=ln(3xy-6)

Q: Determine whether v is in the column space of A, or the null space of A, or both, or neither. -8 -2…

A: First, perform the row reduction to get the row reduction echelon form. Solve Ax=0 to get the…

Q: 3.46. Given below is a proof of a result. What is the result? Proof Assume, without loss of…

A: Here we have given a proof of the results  We assume that x & y are even and z be any integer…

Q: MESE PCca LE Prag the piaen bew onto the fale to the luft se that ah row ntaa funton, i ier fuaston,…

A: This question is based on finding inverse of function.

Q: EXERCISE Show that T = { a+b√3 a,b € Z } is a subring of R

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Q: 1. Let A = (x1, Y1), B = (x2, Y2), and C = (x3, Y3) be distinct points on the plane. Show that the…

A: The determinant is a scalar value that is a function of the entries of a square matrix. It allows…

Q: Topic is Euclident n-space Let x,y,z ∈ Rn , and c ∈ R. Then prove the ff: a. ||x|| >= 0 b. ||x|| = 0…

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Q: 7.)a. Prove that I[x] = {a, + ax+ + ar"|a, = 2k for k E Z}, %3D ... the set of all polynomials in…

A: We need to prove the statements in question 7.

Q: 1- iz If z = x +i y and w = z-i show that |w| = 1-z is purely real

A: i2 = -1

Q: Complete this proof in its proper format: Proof= { (∀x)(∀y)Rxy } |- (∀z)Raz Assumption:…

A: The given statement is ∀x∀yRxy|-∀zRaz. We have to prove this statement in the prescribed proof…

Q: • F2: o 3€ F2 o n,m E F2 = n x m E F2 Prove by structural induction that F C F2

A: given F1→3∈F1→n∈F1⇒n2∈F1 F2→3∈F2→n,m∈F2 ⇒n×m∈F2 to prove by stuctural induction that F1⊂F2

Q: a. Let V be the the set of polynomials of degree less than or equal to 2, with operations defined as…

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Q: Find all values of x and y such that Px(x,y)=0 and Py(x,y)=0 simultaneously.   P(x,y)=…

A: Given function P(x,y)=6x2-12xy+3y2-36x+66y-3 We have to find all values of x and y such that Px=0…

Q: (a) Let p(x) be a polynomial with coefficients in R, and let M abe the 3 x 3 matrix а 1 Ma := a 1 а…

A: Given that p(x) be a polynomial with coefficients in R and Ma be the 3×3 matrix Ma=a100a100a

Q: 8. (The argument principle and Rouché's theorem) (a) Determine the number of zeros in the right…

A: (a) The given function is pz=z3+z2+9z+4 let's consider Pz=hz+qz such that hz=z2+9z+4 and qz=z3…

Q: that Tis a T(x1, X2, X IS

A: (i) Given that, Tx1,x2,x3=x1-5x2+4x3,x2-6x3 Then the standard matrix of T is: 1-5401-6 Domain of T…

Q: 19. Cross and dot products Prove or disprove the formula u X (u X (u X v)) •w = -|u²u•v X w.

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Q: Prove that the dimension of Pn(R), the set of all polynomials of degree less than or equal to n, is…

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Q: Abstract Algebra: Let p(x) = 2x3 −3x2 + 4x−5 and q(x) = 7x3 + 33x−4 in R[x], where R = Z/3Z.…

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Q: Determine whether the following polynomials u, v, w in P(t) are linearly dependent or independent: u…

A: Please refer the attached image for complete solution.

Q: Solution set of en eous write the the fallowing homogensaus aystem in pasmetaic fom fosm Xi +3x2…

A: The system of equation is given as x1+3x2-5x3 = 0x1+4x2-8x3 = 0-3x1-7x2+9x3 = 0 Corresponding…

Q: Determine it A= 0 3 2 diagonalizable. is 3о9 If find X such that X-'AX=D is diag onal matrux. yes,…

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Q: using Cayley - hamilton theorem [1 2 3 Let A = 1 2 Find A5. 0 1 2

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Q: Define injections g : [0, 1) → [0, 1] and h : [0, 1] → [0, 1). Prove that that the functions g and h…

A: We have to show there are two functions, g:[0, 1)→0, 1 and h:0, 1→[0,1) which are one-one…

Q: V = R' = {(x.y)|x, y e R}, with standard operation of addition on R² which is (x,.y,)+(x,, y2) = (x1…

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Q: Given A (w, x, y, z) = (w + x + z)(w' + y)(w' + x' + y')(w + x +y' + z'): a. Express A as a Sum of…

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Q: Heather does a weekly exercise program consisting of cardiovascular work and weight training. Each…

A: According to question,She exercise at least 8 hours,x = Time she spends on cardiovascular work,y =…

Q: f. Both H and J below have their leading 1 in row 3 in the same column: 1 0 0 6. 7 1 0 0 -2 9. H 0 1…

A: Hi, since you have asked multiple questions, we will solve the first question for you. If want any…

Q: Question Let -2 -2 V1 = V2 = 3 V3 = and р — 6. 17 Determine if p is in the columns space of Afv1 v2…

A: The solution is given as

Q: Proof that R3 = W1 ⊕ W2, where W1 = {(x1, x2, x3) : x1 + x2 + x3 = 0} and W2 = Lin(1, 1, 1). Lin -…

A: To given subspaces are W1=x1,x2,x3:x1+x2+x3=0W2=Lin1,1,1 To prove ℝ3=W1⊕W2 In order to prove that…

Q: [1030] Let A = 0 120 %3D V be the null space of A, and W= V.What is the dimension of W? 0 0 00 1 3.…

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Q: %3D Let C be a set in one-dimensional space and let Q(C) = Ec f(x) ,where f(x) = (3/4)(1/4)", x = 0,…

A: Just defining the  Q(c) and then proceed as given..........................

Q: (a) Show that T is linear and bounded, and find ||T||. (b) Show that T is bijective and find T-'.

A: We have to show that T is linear as well as bijective. And we have to find its norm ||T|| as well as…

Q: How many rows and columns must a matrix A have in order to define a mapping from R? into R° by the…

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Q: Proof #1) A Given AC = CB Dis the midpoint of AB Prove: AACD ABCD

A: It is given that, AC=CBD is the midpoint of ABWe have to prove that, △ACD≅△BCD.

Q: Example 6.20. Using the method of separation of symbols, prove that (i) u „x + u,x + ux³ Ди, + A?u,…

A: Here, We need to use method of separation.

Q: Let R1 b = R2 –R3] 1 and -6 0 21 A = 4 -2 7 Determine if b is in column space of A and Null space of…

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Q: Hl9. for a Ux7matrix at least: The dinensiom of the null space is A) 3 0)4c)5 D) 6 E) 7

A: On solving this we will get

Q: 1 0 -1 Let A = Is 6 = 2 in the span of the columns of A? Is w = (-1 1 1) in 1 1 the span of the rows…

A: We need to find whether b is in span of columns of A and w is in span of rows of A.

Q: Let G = Express a3b2abab3 in the form biaj, where 0 <=i <=1 and 0<=j<=3

A: G =&lt;a,b a2=b4=e,ab=b3a&gt;To express: a3b2abab3 in the form of biaj, where 0≤i≤1 and 0≤j≤3

Q: 5. Let W be the space of polynomials p(x) with real coefficients of degree < 10, satisfying p(0)…

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Q: 1) Let P2 = {polynomials f(x) = a2x²+a1x+ao, where ao, a1, a2 E R}, and define addition and scalar…

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Q: 4 7 - 1 Let u = 0 and A = 1 2 -1 Is u in the subset of R° spanned by the columns of A? Why or why…

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Q: Exauple- If v=(T') is a contraviurient vecfor, Shew that He Parhil clenivatokes , dehined in each…

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Q: . The Cauchy-Schwarz inequality says that if a = (a1, ..., an) and b = (b1,... ,bn) are two vectors…

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.