We can define a one-to-one correspondence between the elements of Pn and Rn by p(x) = a1 + a2x + a3x2 + … + anxn-1   (a1, …, an)T = a Show that if p  a and q  b, then (a) αp  αa for any scalarα. (b) p + q  a + b [In general, two vector spaces are said to be isomorphic if their elements can be put into a one-to-one correspondence that is preserved under scalar multiplication and addition as in (a) and (b).]

Q: Duestion : Find the first derivotive of the llowing function. Arcton x2 f(x) = + e (Arccosx)

A: Given    fx=1Arccosx22+eArctanx2 i.e. fx=1cos-1x22+etan-1x2

Q: - c(13, 23, 21, 27, 37, 52, 34, 42, 31, 40, 54, 30, 15, 35, 19, 23, 22, 18) Fine the vector y by the…

A: year x 1951 13 1952 23 1953 21 1954 27 1955 37 1956 52 1957 34 1958 42 1959 31…

Q: 29. Suppose the vectors v, V2, ..., Vr in a vector space V are linearly dependent. Which statement…

A: D is correct option

Q: (c) Show that if a span{b₁,b2, ..., b;} then bib{T bi bi (0.0.3) Hint: You can prove the…

A:

Q: -2x – 4y + 2°z = 1 ) %3D Solve the following linear system. 3x2 + 2x3 – x4 + 2x5 = a 3x1 -9x2 + 7x3…

A: by backward substitution find values

Q: nd the values of the scalars a, b, c such that -2,1, –3) = a(2, -1,1) + b(1,3,–2) + c(-2,1,–3) %3D

A:

Q: Below are 3 sets of 3-dimensional vectors: {}] (i) (ii) {}}} -8 (iii) 3 Determine which set(s)…

A: for the given sets of vectors

Q: Consider the 4-vectors Aª = (a, 0, 2a, 0) and Bª = (23, 0, 0, 0), where a and 3 are non-zero real…

A: and .It is given that  is null vector.

Q: show that the given vector functions are linearly dependent on (−∞,∞).

A:

Q: Can a vector in a vector space R -4 - 2 be 2. span by -1 2 イ

A: Span of vectors.

Q: For any vector A,ix (i x A) +jx (j xA) + kx (k xA) = -2A ylan O İhi O

A: We use vector triple product to answer the given question.

Q: 3- The scalar component of the vector F in the direction of K is (K.F А- .K, В- K C-

A:

Q: If vectors b, c and d are not coplanc then prove that vector

A:

Q: Find all vectors in R* that are orthogonal to both 1 1 V1 = and V2 2 -2

A: Linear Algebra: The vectors are: v1=112-2v2=2112Let denote the vector v3=v1v2v3v4

Q: (1,1,..., 1)T be the column vector consisting of all 1s. Prove 0. 3. Let a = that La

A: Given: Let a=(1,1,......1)T  Then objective is to prove  La=0

Q: Show that in the vector space R³ the vectors x = (1, 2, -1), y = (3, 1, 1), and z = (5, -5,7) are…

A:

Q: Consider the set V = R − {2}, that is, the set of all real numbers except 2. Define “addition” as…

A:

Q: Determine the basis that generates the following vector space by clearing the variable y: M = {(x,…

A:

Q: Let the following vectors be in R³ a = (1,3,-1) b= (-2,1,3) c = (3,2,1) d = (1,-1,2) 16.- Obtain a…

A: Given vectors are a=(1,3,-1), b=(-2,1,3), c=(3,2,1), and d=(1,-1,2).

Q: Is v = |1a vector in the span of S = }|2| Use Gaussian elimination to support your answer.

A: To check v=111 a vector in a span of S=123, 012, -201 or not.

Q: Q1. (a). Let V = R and define addition and scalar multiplication by: (a, b) e (c, d) = (a +2c,…

A: See the detailed solution below.

Q: 9. Show that the following statements about two vectors (a, b), (c,d) = R² are equiv- alent. (a) ad…

A:

Q: Suppose v1, ..., vn are linearly independent and Span({v1, .., vn}) = V , then if W ⊂ V and W…

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Q: Consider the set V of all real triples of the form (a, b, c) where c = a - 2b under the usual vector…

A:

Q: Use Exercise 47 to show that if cos α, cos β, and cos γ are the direction cosines of a vector v,…

A: cos2α,cos2β and cos2γ are the direction cosines of the vector v

Q: Let E be the solid bounded by X = 0, y = 0, z = 0, x = 1, 4y + 2z = 8. Evaluate E 2 8 4 1 x dV. None…

A:

Q: 1 Defined in P,(R), p(x)=1,q(x)=x- and h(x) =x² -: -x+- 6. Investigate whether the vectors are…

A:

Q: Let x1(t), x2(t), . . .  , xn (t) be vector functions whose ith components (for some fixed i )…

A: Given that x1(t), x2(t), . . ., xn(t) be vector functions whose ith components ( for some fixed i)…

Q: Determine whether or not the following set of vectors span R³, (i) {(1, 1, 2), (1, -1, 2), (1, 0,…

A: Span of R3 .

Q: -0-0--0--0--0- 1 19. Let S be a set of vectors such that u = whether the set of vectors is linearly…

A:

Q: If a, and b are non - collinear vector and vectors a = (x +4 y)a +(2x + y +1)b and, B= (-2x + y + 2…

A: We will solve the following

Q: 2 Consider A = -2 null(A). 4 2 2 4 -4 -2 -2 -4 -2 0 1 Give a set of vectors that spans each of im(A)…

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Q: Let V1 MECANIS 3 1 11 V2 = 1 0 V3 = SCAMENTO 0

A:

Q: r projection oE the following

A: we've to show whether the projection operators are positive operators or not. Now a projection…

Q: or function By – k1 z) + ay (kz y + z) onal, then the values

A:

Q: 2. Consider the collection of vectors --0·00) 2 S = Is S linearly independent or linearly dependent?…

A: