Explain how fixing a basis v1, , Vn of V associates to a quadratic form a corresponding function in n variables of the form q(x1,· · · , Xn) = Eij=1 aijX;Xj: for a certain symmetric matrix A = (aji). ...

Q: Find the coordinates of the vector v = (7,4) with respect to the basis U₁ = (2, 1) and u₂ = (3,1).

A: The answer is given below:

Q: 2. Find an orthonormal basis for the image (column space) of A, where 1 2 3 A = 2 4 6 3 6 0

A: A=123246360  ~ 12300000-9 R2→R2-2R1R3→R3-3R1 ∴it for only first and 3rd column leading coefficient…

Q: Prove that the vectors (1. 1. 0) (1 2 3) and (2. –1 5) form a basis for R3

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Q: If {u, v} is a basis for a vector space V, show that {u – 2v, au} is also a basis for V for every…

A: Here we use the definition of basis and linear combination of vectors as follows

Q: Find an orthonormal basis of ℋ:  x− 2y − 3z = 0.

A: Introduction: The Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner…

Q: i) Find the matrix of T with respect to ordered basis B' = {(1, 1), (1, – 1);:.

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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: Prove that the row vectors of an nxn invertible matrix A form a basis for R".

A: Rows of A form a basis

Q: Let Find an orthonormal basis of the row space of A. A = 1 4 1.[C ]]}.

A:

Q: Find the missing coordinates such that the three vectors form an orthonormal basis for R3 : -0.8 0.6…

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Q: Let {u1, u2, u3} be an orthonormal basis for an inner product space V. If v = au1 + bu2 + cu3 is so…

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Q: 18. Use the Gram-Schmidt process to transform the given basis (w, w, orthogonal basis where w =…

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Q: Let {u1, u2, u3} be an orthonormal basis for an innerproduct space V. If x = c1u1 + c2u2 + c3u3 isa…

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Q: Let B be the basis of R? consisting of the vectors 4 -2 and let C be the basis consisting of 3 2…

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Q: Find a basis of R3 that contains the vectors (1,0,1) and (1,-1,1)

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Q: Find a basis for the set of vectors in ℝ2 on the line y = 5x.

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Q: Q1:Find an orthonormal basis for the Euclidean space R* that contains the vectors U1 uz = and uz

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Q: Find a basis of R that contains the vectors X = (1,1,1,1) and Y = (1,1,0,1).

A: Given Vectors X =(1,1,1,1) and Y = (1,1,0,1) Now determine the orthogonal basis for R4

Q: Verify that the polynomials 2 – x², x' – x, 2 – 3x² and 3 – x' form a basis for P3(x). Express cach…

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Q: (c) Show that the matrices M₁ vector space M22 of 2 x 2matrices. M₂= M₁ = = 1 form a basis for the

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

Q: The Gram-Schmidt process transforms the basis u,=(1, –3), u2=(-7, 1) of R- into the orthogonal basis…

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Q: Let V = [- 7, – 7,0] and V½ = [0, 3, – 4]. Use the Gram- | Schmidt procedure to construct an ordered…

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Q: Find a basis for R2 that includes the vector (2, 2).

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Q: The Gram-Schmidt process transforms the basis u,=(1, 3), u,=(1, –7) of R² into the orthogonal basis…

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Q: 3) Let C be a basis that contains vectors (1 + (-1) x + 1x² + 1x³), (1 + 2x + 2x² + (−1) x³), (1 +…

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Q: Let Find an orthonormal basis of the column space of A. A = 1 1 -3 0 -2 -3 1

A: Let A=1-301-2-3-116   To Find: An orthonormal basis of the column space of A.

Q: Find a basis for the solution space. (If a basis does not exist, enter DNE into any cell.) X1 - x2 +…

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Q: Q1:Find an orthonormal basis for the Euclidean space R* that contains the vectors 1 and uz = ; uz =…

A: Now, from the grem scehem process we have:

Q: (a) Find the coordinate vector of æ with respect to the ordered basis E = -2 of R3:

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Q: If the vectorsV1,..., Vk form an orthonormal basis of OC", then they also form a basis of R*. O True…

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Q: Find an prthonormal basis of the plane x1+ 2x2- x3 0.. W

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

Q: Find an orthonormal basis of the plane x1 + 8x2 – x3 = 0.

A: That's easy. Have a great day!!!

Q: Consider the matrix -1 A = 1 1 Find a basis of the orthogonal complement of the null space of A. ||

A: We know that Null Space of A={ x∈ℝ2 : Ax=0 }where x=( x1 , x2 )Orthogonal complement of vector  x=(…

Q: Prove that the vectors (1, 1, 0),(1, 2, 3) and (2, −1, 5) form a basis for R3

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Q: 3. In the space of square matrix M2x2 find representation of vector A = L4 from the basis U1 ; Uz =

A: As basis is given . So, vector A can be written as linear combination of basis vectors .

Q: If f(x) = x^2 + 7x+2, g(x) = x^2 - x + 1 find a polynomial h(x) of degree 2 such that together these…

A: Given:  f(x)=x2+7x+2,g(x)=x2-x+1 We have to find a polynomial hx of degree two such that together,…

Q: The Gram-Schmidt process transforms the basis u3(1,-3), u,=(-7, 1) of R into the orthogonal basis…

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Q: A is a 3 x 5 matrix and (v1, v2) is a basis of null(A). The vectors v3, v4, vz E R³ satisfy Avz =…

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Q: Let {v1, ... , vn} be an orthogonal basis for Rn and let W = span(v1, ... , vk)· Is it necessarily…

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Q: Let {u₁, U2, u2} be an orthonormal basis for an inner product space V. Suppose v = au₁ + bu₂ + cuz…

A: We have v=au1+bu2 +cu3 Now, ∥v∥=a2+b2+c2 Given ∥v∥=80 So,  80=a2+b2+c2⇒802=a2+b2+c2⇒80=a2+b2+c2…

Q: The Gram-Schmidt process transforms the basis u=(1,2), u2=(2,-6) of R into the orthogonal basis…

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Q: In the vector space f, of polynomials of degree at most two, the third column of the matrix of…

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Q: Let Find an orthonormal basis of the column space of A. 2 5 3 -5 3 2 us

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Q: Let P2(x) be the vector space of polynomials with real coefficients of degree at most 2 with the or-…

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Q: Find the coordinates of x = {-10, 2, 4} with respect to the basis consisting of u, {1, 3, 2}, u, =…

A: Find {a, b, c}.

Q: Find a standard basis vector that can be added to the set {V1, v½} to produce a basis for R а. 0 —…

A: We have to solve given problem:

Q: The Gran-Schmidt process t ransforns the basis u,-(-3,1), uz-(7, 1) of R into the orthogonal basis…

A:

Q: In the vector space P, of polynomials of degree at most two, the coordinates of the vector…

A: The basis of the given polynomial is: B= e1=1+2x,  e2=x-3x2,  e3=1+x2

Q: 9. Let i, j denote the standard basis in R2. Find a scalar c if the vector 7i- 2j is perpendicular…

A: Dot product of the two vectors.

Q: 3. (а) Find an orthonormal basis B = {ū1, ū2, ū3} of R³ such that ủi and the x-component of ūz is…

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.