Q2 Consider the orthogonal basis 1 1-i v1 = i V3 = 1+i -1+i of the inner product space C with the usual inner product. Find the unique vector u e C® such that (u, v:) = 1 for i = 1,2,3.

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: Let V be a vector space, let c be a scalar and let u, v, w be vectors in V and ca scalar. Which of…

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Q: Determine whether the vectors (3,0, 3, 6), (6,0, 0, 3), (0,0,3,0),(6,3,9,12) form a basis of R*.

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Q: (a) Give examples of three distinct vectors v1, v2, v3 in the vector space R3 such that v2 E…

A: Let v1 = {1,1,1}v2 = {2,3,4}v3 = {1,2,3} Here v2 = {2,3,4} = {1,1,1}+{1,2,3} = v1+v3

Q: 2. fu = and v = v1, v2, V3 > are two vectors in 3-space show k that u x v= u1 U2 U3 V1 V2 V3

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Q: EXAΜPLE2 Determine whether the vectors vi-(1,-2,3),v2=(5,6,-1),v3-(3,2,1) are linearly independent…

A: Given vectors are v1 = (1, -2, 3), v2 = (5, 6, -1), v3 = (3, 2, 1)

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

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Q: Determine a basis for the vector space spanned by the vectors 3. V1 2 ,V2 .Vs 17 12 O A. {V1, V3,…

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Q: Let W be the set of all vectors of the form where a and b are real numbers. 2a+b (a) Determine…

A: Given, W=ab2a+b where a and b are real numbers.

Q: Let V1, V2, V3 be vectors in R³ given by 1 »-(;) · ~ -(:). = 6 V2 = 2 2 0 The set {V1, V2, V3} is…

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Q: Let {u₁, U₂, U3, U4} be the orthogonal basis for R¹ given below. Write x as a sum of two vectors v₁…

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Q: If L is a line in 2-space or 3-space that passes through the points A and B, then the distance from…

A: Given, A=( 1,1,0 ); B=( -2,3,-4 ); P=( -3,1,6 )To find: Distance from point P to the line through A…

Q: Using the transformation law for vectors, prove that for any two vectors Ā, B, the inner product of…

A: As per the question we have to prove that the inner product of two vectors is invariant of…

Q: Given any linearly independent set of vectors V = {v, V2, , }, prove that there is at least one…

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Q: Find the projection of vector u⃗ = −i + 5j + 3k onto the vector v = −i + j − 3k?

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Q: Use matrix multiplication to find the orthogonal projection of (-8, 5, 7) on the xy-plane. Let T…

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Q: Let B' = {u1,u2} and ß = {v1,v2} be ordered bases for a 2-dimensional vector space V. Given that [u1…

A: Given u1+u2β=5,4t, and the change of coordinate matrix Q from β' to β is Q=1c3d. Express v1 as a…

Q: Find an orthonormal basis of the plane x₁ +4x2x3 = 0.

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Q: Find an orthonormal basis of the plane x1+ 5x2 – x3 = 0. }

A: Introduction: A set of vectors that is orthogonal to each other and spans a plane is the set of…

Q: Let Define addition and scalar multiplication as follows. b +e) Ic+f kby verify the following…

A: Given vector is: V=1bc1 b, c∈RThe addition and scalar multiplication are defined…

Q: Let v1=( 0 1 -1 2)^T v2=(1 0 -2 1)^T Let V=Span{v1,v2} Find a basis for the orthogonal complement of…

A: Orthogonal complement : Let V be a vector space in ℝn. The orthogonal complement is the subspace…

Q: I) Determine whether S is a basis for the indicated vector space : 1) S = {(3,-2), (4,5)}for R² 2) S…

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Q: y" + 9n²y = 3ñd(t – 4), a. Find the Laplace transform of the solution. Y(s) = L{y(t)} b. Obtain the…

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Q: Do the vectors form a basis for R"? Show which conditions, if any, are met, and show which…

A: First one not form a basis but second one is.. See the step 2

Q: Consider the vectors V1 = U₁ = v2 = and v3 = Convert the set {V₁, V2, V3} into an orthonormal set…

A: Given vectors are  v1 = 451 ,  v2 = -32-6 ,  v3 = 16-7-23 given inner product is  u, v = 2u1v1 +…

Q: nd scalars a, b and c such that a(2,0,1) +b (1,3,2)+c(1,0,1)=(7,3,4)

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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.