Q#3: Let V= P(R), and for P) EV, defme f,f EV by f(PW)= PE)dt and (Po)= Pe) dt. Find a basis for V for which it is the dual basis.

Q: Explain why S is not a basis for R2. S = {(-3, 6)} O sis linearly dependent. O s does not span R². O…

A: We have to find out

Q: Let B = {v1,..., Vn} be an orthonormal basis for R". (a) Prove Parseval's Identity, that is, for any…

A: According to the given information,

Q: Q1:Find an orthonormal basis for the Euclidean space Rª that contains the vectors -HH [0] u1 = ; Uz…

A: Here we have to find an orthonormal basis for the Euclidean space R4 that contains the vector…

Q: Let V = P2(R), give V the inner product defined by (f,g) = Lo s(1)g(t) dt, and let W = span {1 – x}.…

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Q: Explain why S is not a basis for P2. S = {1, 4x, -6 + x², 7x} O sis linearly dependent. O s does not…

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Q: Explain why S is not a basis for P. S = {1, 8x, -4 +x², 9x} O sis linearly dependent. O s does not…

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Q: 4. Apply the principal axis transformation to the curve of second order -x2 + 6xy – y? – 22.x+ 18y –…

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Q: Explain why S is not a basis for P. S = {1 – x, 1 – x², -1 – 2x + 3x?} %3D O S is linearly…

A: We will check if the vectors given in set S are linearly independent or dependent also determine if…

Q: Q2) (a) Use the definition = ]f(t)g(t)dt in T0,1] to find the components of f(t)=2+t in the…

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Q: Explain why S is not a basis for R². S = {(-6, 2)} OS is linearly dependent. S does not span R². OS…

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Q: Explain why S is not a basis for P2. S = {1, 2x, –4 + x², 3x} O sis linearly dependent. O s does not…

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Q: Explain why S is not a basis for P2. S = {1, 4x, -9 + x², 7x} S is linearly dependent. O s does not…

A: Given S={1,4x,-9+x2,7x}. We explain why S is not a basis for P2.

Q: Explain why S is not a basis for R2. S= {(6, 5), (24, 20)} O sis linearly dependent. O s does not…

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Q: Prove that the components of a vector x in a given basis {ê¿} are unique. That is, show that if you…

A: We will prove the given statement.

Q: Let S = y1 be a basis of R and x = E R4. What are the coordinates of x with %3D >Y2 = respect to the…

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Q: Explain why S is not a basis for R². S = {(2, 1), (4, 2)} O Sis linearly dependent. S does not span…

A: Let S=2,1,4,2 Since if S is basis of R2 then S is linearly independent and S spans R2 . Let S is…

Q: Explain why S is not a basis for P2. S = {1 – x, 1 – x², –1 – 6x + 7x²} S is linearly dependent. O s…

A: Consider a matrix formed by vectors of S and determine the determinant of the matrix, 1-1010-1-1-67=…

Q: (11) Find the dual basis {V4.V } of the stan dard basis of IR. Let T:IR → IR defined by T(xiy, z)…

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Q: Explain why S is not a basis for R². S = {(-5, 7)} O Sis linearly dependent. O s does not span R2. O…

A: use the definition of linear independent set. and of basis.

Q: Explain why S is not a basis for P2. S = {1, 7x, -6 + x², 2x} O sis linearly dependent. O s does not…

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Q: Define T:R?→R? by T(x) = Ax. Find a basis B for R with the property that [T]B is diagonal. 2 - 4 A =…

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Q: {( a + b+c+d b – c+ 2d а + 2с — d H = : a, b, c, d e R } C R³, and consider --(}) u = 1 (а) Is u E…

A: Given, H=a+b+c+db-c+2da+2c-d: a,b,c,d∈ℝ⊆ℝ3      ⋯⋯(1) a) Given, u=111 Consider the system,…

Q: Explain why S is not a basis for R. S = {(-6, 4)} O s is linearly dependent. O s does ot span R. O s…

A: We will explain the reason.

Q: Let P be a plane in R3 defined by the equation 3x + 2y − z = 0. (a) Find an orthonormal basis for…

A: To find the orthonormal basis we first find basis for P and then use Gram-Schmidt process.

Q: Explain why S is not a basis for R2. S = {(-5, 4)} O is linearly dependent. O s does not span R2. O…

A: The given set,  S=-5,4

Q: Explain why S is not a basis for R2. S = {(-5, 2), (o, 0)} O sis linearly dependent. S does not span…

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Q: Given T : R2 --> R2 defined by T (x, y) = (x - 2y,  x + y) (a) Is T is injecting? (b) Determine…

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Q: Explain why S is not a basis for M, 2: -{::}[::} 0 1 1 S = 0 1 1 0 S is linearly dependent. s does…

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Q: Explain why S is not a basis for R2. S = {(-6, 4)} O sis linearly dependent. O s does not span R2. O…

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Q: Explain why S is not a basis for P2. S = {1 – x, 1 – x², –1 – 3x + 4x²} O s is linearly dependent. O…

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Q: 45. Let {v, w} be an orthonormal basis for R², and let T: R2 → R² be the function defined by T(u) =…

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Q: Explain why S is not a basis for P2. S = {1 – x, 1 – x², –1 – 4x + 5x²} O S is linearly dependent. O…

A: Consider the given functions. S=1-x,1-x2,-1-4x+5x2 Let S is linearly independent and there exist…

Q: Explain why S is not a basis for R2. S = {(-6, 6)} O sis linearly dependent. S does not span R². S…

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Q: ow that B = {5+i, 2+i} is a basis for Q(i) whose elements are algebraic tegers. Why is ß not an…

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Q: Explain why S i not a basis for R². S = {(-3, 7)} O sis linearly dependent. O s does not span R2. O…

A: We will find out the required solution.

Q: Explain why S is not a basis for P2. S = {1, 2x, -5 + x², 9x} O sis linearly dependent. O s does not…

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Q: Explain why S is not a basis for R2. S = {(-3, 4)} S is linearly dependent. S does not span R2. O…

A: The set S=-3,4 is not a basis for R2.

Q: Explain why S is not a basis for R2. S - {(-5, 2)} O sis linearly dependent. O s does not span R2. O…

A: Basis :  The set of vectors v1,v2,...,vn ⊆V in a vector space V is called a basis for V if (i) S is…

Q: If {v1,..., Vn} is an orthonormal basis for R", then vịvi + · · ·+ vnvn = I. O True O False

A: Given, v1, v2, ..., vn is an orthogonal basis for Rn. then v1v1T+...+vnvnT=I It is True or False. To…

Q: Explain why S is not a basis for R2. S = {(2, 5), (8, 20)} O Sis linearly dependent. O s does not…

A: Let's find.

Q: Explain why S is not a basis for R2. S = {(-4, 2)} O S is linearly dependent. O s does not span R2.…

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Q: Explain why S is not a basis for P. S = {1, 2x, =7 + x², 9x} O S is linearly dependent.

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Q: 7. Consider the basis B-{v,,Vz} for IR? where Vi =(I,1) and Vz =(-1,1). Let T:IRIR? oe a linear…

A: NOTE: Refresh your page if you can't see any equations. .

Q: Explain why S is not a basis for R². S = {(-3, 7), (0, 0)} O sis linearly dependent. O s does not…

A: Let V=v1,v2∈R2.

Q: B = (f1, f2; f3) is a basis for R2[r] where fi = 1+z+2r², f2 = 1+2r + 5x², f3 = 2+1+2r². If g e…

A: Coefficient of linear combination is given. We have to find the element

Q: Explain why S is not a basis for R2. S = {(-4, 2)} O sis linearly dependent. Os does not span R. S…

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Q: Define T : P2 → R² via T(a + bx + cx²) = (a + b, b – c) . Find a basis for ker T. Also, find dim(ker…

A: The detailed solution is as follows below:

Q: Explain why S is not a basis for P₂. S = {1, 6x, -2 + x²,4x} O S is linearly dependent. OS does not…

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Q: |– 1/V5 29) Let u,= 2/ 5 2/V5 [1/v5] Show that {u1,u,} is an orthonormal basis for R².

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Q: 10)Define TE L(R²) by T(x, y) = (41x + 7y,-20x + 74y). Verify that the basis of T %3D with respect…

A: Question 10. The detailed solution is as follows below: