Let f(x) = 6, g(x)= 8x + 7 and h(x) = -3x² + 2x + 4. Consider the inner product (p, q) = p(-1)q(-1)+ p(0)q(0) + p(1)q(1) in the vector space P, of polynomials of degree at most 2. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of P, spanned by the polynomials f(x), g(x) and h(x).

Q: Let A - ( ) a. Show that the vectors lz.A, A² are linearly dependent in Maz (R). b. Deduce that A"…

A: Statement (S1) proves that  I2 , A, A2  are linearly dependent.

Q: Determine a scalar s such that A2x = sx when Ax = rx.

A: To determine a scalar s such that A2x=sx when Ax=rx Given Ax=rx On multiplying both the sides by…

Q: 3. Let W = M2x2(C) with zero determinant. Show that W is not a subspace of V = M2x2(C).

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Q: Let Q(x) = 7x} +x} +7x} – 8x1x2 – 4x1x3 – 8x2x3. Find a unit vector x in R3 at which Q(x) is…

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Q: Find the point on the plane 2x + 3y + 4z = 0 which is closest to the point (1,0, 4) in two %3D…

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Q: Let M2x2 be the vector space of real 2 x 2 matrices with entrywise addition and scalar…

A: If 0 is an eigen value of matrix A then it is singular.  Since det(A) is equal to product of eigen…

Q: Let {u1 (x) = -9, u2 (x) = –12x, Uz3 (x) = –12x²} be a basis for a subspace of P2. Use the Gram-…

A: we have goven basis of P2B=u,(x)=-9,u2(x)=-12x,u3(x)=-12x2f,g∫01f(x) g9x0dx   on c[0.1]According to…

Q: Prove that if {y1,...,yn} is an orthogonal set of nonzero vectors, then the vectors {x1,...,xn}…

A: Let consider the orthonormal set of nonzero vector be {w1, w2 …wn} instead of {y1...,yn} and the…

Q: Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned…

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Q: Let V be an m xn matrix with orthonormal columns, and x , y be in R" then prove that the linear…

A:

Q: What is the relationship between the kernel of a linear transformation from a finite dimensional…

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Q: Consider the vector space P, of all polynomials of degree = a, a,+b, b2 , then the inner-product of…

A: Given polynomials are: p(x)=a1+b1xq(x)=a2+b2(x) Inner product function is defined as:…

Q: . Show that (x, y) = 4x1y1 – 7x2y2 for vectors x = (x1, x2) and y = (y1 y2) in R2 defines an inner…

A: Given two vectors x and y in R2. We need to show that the given function in x and y defines an inner…

Q: suppose Vn is the vector space of polynomials of degree <n-1 and with real cofficients. Determine…

A: As per the question we are given a vector space Vn of polynomials with degree ≤ n-1 and with real…

Q: Find an orthonormal basis for the subspace of Euclidean 3-space below. W = {(x1, x2, x3): x1 + x2 +…

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Q: .

A: Given that, The vector space C[-1, 1] with an inner product defined by, A set of vectors form an…

Q: Prove that an m X n matrix A has rank 1 if and only if A can be written as the outer product uvT of…

A: Given that an m X n matrix A has rank 1 if and only if A can be written as the outer product uvT of…

Q: (a) Find an orthonormal basis of the vector space P₂ of polynomials of degree at most 2 with respect…

A:

Q: Find the matrix of the standard inner product on W (obtained by restriction of the dot product on…

A: “Since you have asked multiple questions, we will solve the first question specified by you i.e.…

Q: Let T be a linear operator on a vector space with a T-invariant subspace W.Prove that if x is an…

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Q: Consider the vector space 2 with the inner product rgk)=x+1,then %3D

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Q: 3 Find a basis of the subspace of R' defined by the equation -7x1 – 5x2 – 4x3 = 0.

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Q: Consider the vector space P, of all polynomials of degree = a, az+b, b2, then the inner-product of…

A: If p(x)=a1+b1x and q(x)=a2+b2xThen&lt;p,q&gt;=a1a2+b1b2

Q: Find vectors x and y in R2 that are orthonormal with respect to the inner product (u, v) = 3u₁v₁ +…

A: The given problem is related with inner product. We have to find the vectors x and y in ℝ2 that are…

Q: Now assume that f and g are both C2. Express the matrix of second derivatives V²F(x) in terms of f…

A: b)

Q: Let V be a finite-dimensional vector space over F, and suppose that T in L(V, V) has the property…

A:

Q: Suppose that you have three vectors: f1 (x) = 1, f2 (x) = x – 1, and f3 (x) = ; (x² – 4x + 2), that…

A: Suppose that you have three vectors: f1(x)=1, f2(x)=x-1, and f3(x)=12x2-4x+2, that make up an…

Q: Let M1 = and M2 = Consider the inner product (A, B) = trace(ATB) in the vector space R2x2 of 2 x 2…

A: The solution is given as

Q: Suppose V is a 4-dimensional vector space, and that T is a linear operator on V with minimal…

A: Dimension of Ker T

Q: give an example of an infinite dimensional vector space V over complex numbers and a linear operator…

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Q: Is W a subspace of the vector space? W is the set of all matrices in Mn,n with zero determinants.

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Q: Consider the vector space P, of all polynomials of degree = a, a2+b, b2 , then the inner-product of…

A: We have to find the inner product of the following -  p = 2 + 1.7x q = 4.8 + 3.2x

Q: Determine a scalar s such that A2 x = sx when Ax = r x.

A: Given that Ax=rx. Multiply both sides of the above equation by A and simplify as follows.

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 A be an m x n matrix with orthonormal columns, and x, y be in R" then prove tat the linear…

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Q: Compute the standard matrix of the linear operator M : R³ → R° that first reflects all vectors about…

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Q: Show that the set of all polynomials of degree at most 3 forms a subspace of Pn.

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Q: .

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Q: Suppose A is invertible and orthogonally diagonalizable. Explain why A–1 is also orthogonally…

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Q: Find a 3 x 3 matrix A (with real number entries) whose characteristic polynomial is det(A – rls): *…

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Q: Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned…

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Q: Consider the vector space P3 of polynomials of degree at most 3 together with the zero polynomial..…

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Q: Find the standard matrix of the projection of a vector in R³ to the plane – 2x – 5y + z = 0. 0.8667…

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Q: Consider the vector space P, of all polynomials of degree = a, a2+b, b2 , then the magnitude of…

A: Given that the vector space is set of all polynomials of degree less than or equal to 1. For…

Q: Use the Gram-Schmidt process to find an orthonormal basis for im(A), where 2 1] 1 1 -2 0

A:

Q: Let f(x) = -1, g(x) = -7x + 2 and h(x) = 5x². %3D Consider the inner product (P, 9) = / P(=)q(=) dz…

A: Given fx=−1, gx=−7x+2 and hx=5x2 And p,q=∫04pxqxdx Gram Schmidt orthogonalization process is  ax=fx…

Q: Suppose that the deformation of a continuous body is given by 1 x(X) = (3 – 2X1 – X2)e1 + (2 +;X1 -…

A: The deformation of a continuous body is given by χX=3-2X1-X2e1+2+12X1-12X2e2+X3e3 The matrix…

Q: Explain how fixing a basis v1, , Vn of V associates to a quadratic form a corresponding function in…

A: To explain how a basis v1,v2,...,vn associates to a quadratic form of a corresponding function in n…

Q: 2. Let S = {vi, v2, V3} be a set of linearly dependent vectors in an inner product space V. What is…

A: To explain: What will be the result when we apply Gram-Schmidt process to a linearly dependent set…

Q: x=y=z

A:

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.