If Pa is the Vector Space of poly romiels Of degee ss and Be is tho kctor of Spaco Of purynomals of degree I2. Prove that Pais a subspace of Pz ,find a basis for the Sub Space,

Q: (a) {(a, b, c) ER³ | 2a = 3c}. (b) {(a, b, c) ER3 | a = 0, b = 3c}. (c) {(a+b, ab, 2a + b) | a, b R}…

A: (a), (b) , (c), (d)  are subspaces .

Q: 4. For vectors v and ū in an inner product space V, prove that u´ projzu and i are orthogonal.

A: v

Q: A set of 4 vectors in R5 is linearly independent cannot span R5 spans a 4-dimensional subspace of…

A: Given a set of 4 vectors in ℝ5.

Q: 2. Let a, B, y be three vectors in the inner product space V½ (R) with the standard inner product…

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Q: Find llFll and sketch several representative vectors in the vector field.

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Q: If a vector space V has dimension 5, and we have a subset {v1, v2, ., Us} CV, can S be linearly…

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Q: 27. Let S be the set of all vectors both of y in R' such that x + y -z = 0 and 2x – 3y + 2z = 0.…

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Q: Let V = Poly2 be the vector space of polynomials of degree 2 or less and let W = Poly, be the vector…

A: Suppose we have given a linear transformation ,   is basis of V ,  is basis of W  and associated…

Q: 1) Let v be an element of a vector space V. Prove that -(-v) = v. Your proof should work in any…

A: Here a,b are element of field.

Q: Suppose W,, W2,. (a) span (W, W2 (b) If S; spans W; for i = 1,...,r, then S, US, U...US, spans W, +…

A: (a) W1+W2W1+W2 would be W2W2. Below is the justification.  x+y∈W1+W2,wx+y∈W1+W2where x∈W1x∈W1 and…

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A: We have to determine if the vector spaces, are exactly the same or there is any similarity or…

Q: W4 Pick any vector space V that you want, as long as dim(V) ≥ 6. Then, pick 4 vectors in the space,…

A:

Q: , dim W 5. Let U, W be subspaces of a vector space V. If dim U and Un W = {0}, what is dim (U+ W)?

A: Let V be a vector space over a field k and let U, W be finite-dimensional subspaces of V.  Then…

Q: 14. What are the conditions on a set of vectors S to be a basis for a vector space V?

A: If V is any vector space and  S = {v1, v2, ...... Vn} is a set of vectors in V, then the set S is a…

Q: Show that the function (V1, V₂) = 2w1W2 + x1x2 - Y1Y2 +2122 defined for any vectors V₁ = (x1, y1,…

A:

Q: Suppose V,W are subspaces of R^n. Let V ∩ W be all vectors which are in both V and W. Show that V ∩W…

A: Suppose V,W are subspaces of R^n. Let V ∩ W be all vectors which are in both V and W.  We have to…

Q: ra Which of the following is a basis for the subspace U = O -08 a) -88 d) e) HB 88 c) - 2b = c } <R²…

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Q: Is the following set of vectors S = {v,, V2. V3} basis for R³ space? Where, a. v = (3,1,–4), v2 =…

A:

Q: Recall for subspaces U, V of R",U +V = {u+v : u e U, v e V}. This is a subspace of R" referred to as…

A: The proof of given problem is given as follows :

Q: Suppose V is finite-dimensional and U and W are subspaces of V with W^0 ⊂ U^0. Prove that U ⊂ W.

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Q: d. Find the transition matrix from C to B. e. Find the coordinates of u = (2, –2) in the ordered…

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Q: Any nonzero subspace W C R" has an orthonormal basis. True O False

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Q: If vector space V has dimension n, then any subset of m <n vectors can span V. Select one: O True…

A: Statement:- If vector space V has dimension n, then any subset of m&lt;n vectors can span V.

Q: 2. A 2D-subspace (manifold) in 3D- space is defined by the two vectors d;=é,+ê,-2êzand ā,=ê,+2êz-3ê,…

A: A 2D-subspace (manifold) in 3D - space is defined by the two vectors a→1=e^1+e^2-2e^3 and…

Q: Show that Null Space of Ax = 0 is a subspace. To get full credit you need to show that 1.0 is in the…

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Q: Prove that if IS or thogonal to both WaXhen is orthe gonal to every it' vector 1> 1X

A:

Q: Let (V,(*,*)) be an F-inner product space,where F is either R or C. LetU C V be a subspace and let a…

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Q: Let V be an F vector space of dimension n. Prove that, for k≤n the vectors V₁, V2, …, Vk are…

A: A set of vectors {v1,v2,...,vk} is linearly independent if the vector equation x1v1+x2v2+···+xkvk=0…

Q: Suppose we have n> 1 vectors in R" vector space where the ith element of the ith vector (for 1 ≤ i…

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Q: Let P2 be the vector space of all polynomials of degree ≤ 2 with coefficients in R, and S= {1 + 2x,…

A: The basis for the vector space P2 is 1, x, x2. We have to write the matrix form for the set S. We…

Q: Let W be the subspace of IR' spanned by the vectors 1 4 -2 and -1 -7 Find the matrix A of the…

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Q: Any given vector space may have distinct inner products.  True  False.

A: Given :- Any given vector space may have distinct inner products. To find :- The given statement is…

Q: Let P3 is the vector space of polynomials of the degrees3,Consider the polynomials…

A: Given :  P3 is the vector space of polynomials of degree less than or equal to 3. To prove :  We…

Q: 1.W.4 We'll work inside the vector space of polynomials in degree < 2, which is denoted P<2. Let P1…

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Q: Show that the set of polynomials of degree 3 form a vector space. P(x) = ax³ + bx² + cx+d, a, b, c,…

A: We can show that the set of all polynomials of degree 3, where P(x) = x^3 + bx^2 + cx + d (a, b, c,…