Find an orthogonal basis for ℝ4 that contains the following vectors. [1 4 −1 0] and [1 0 1 3]. Hint set x3 = [0,0,1,0] and x4 = [0,0,0,1] and apply the Gramm-Schmidt Process.

Q: Show that the vectors (3+ /2,1+V2) and (7, 1 +2\/2) in R² are linear!ly dependent over R but…

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Q: 1. Let u = [1 0 1]",0₁ = [1 1+ T -i]¹, 1+i 1- - V₂ = [2i 1 0], and v3 V3 = [2 −1 1]². Let = V1, V2,…

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Q: (a) Prove that the vectors C = { 0 (c) Let B = { 2 } are a basis of R³. (b) Find the coordinates of…

A: Theorem 1: Let V be a vector space of dimension n over a field F. Then any linearly independent set…

Q: Dases C= and E on R3, (i) Find the vector v3 of R' whose component vector with respect to C is [v3]c…

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Q: Let B = vector of 7 = E [V]B 0 1 00 {[%] [83] [28] [89]} 9 9 relative to B. -5 -2 || be an ordered…

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Q: Consider the 4-vectors Aª = (a, 0, 2a, 0) and Bª = (23, 0, 0, 0), where a and 3 are non-zero real…

A: and .It is given that  is null vector.

Q: 5. Consider the vectors V₁ = •[1-* a. Express x = and V₂ = 2 which are orthogonal vectors. as a…

A: 5. Given that the vectors , which are orthogonal vectors.a. To express as a linear combination of

Q: Find Q for the QR factorization of A = = Ex: 1.23 3 -A V1 = 0 , V₂ = 3 10 -9 -3 0 0 10 1 -10 0.21 ,…

A: Given:Also the orthogonal vectors are  We have to find Q for the QR factorisation of given matrix A…

Q: 4). Find all values of a for which the given vectors form basis for R' -(:) --() ---() a? u = and w…

A: A set B is said to be basis of any vector space if (1) it is linearly independent set. (2)B spans…

Q: Let TilP > Pz, the basis of NIT) T (ax²+bx+c) = 2Qx+ b. Then %3D is

A: Applying nullity definition we have

Q: 5. Determine whether or not the following vectors i R³ are linearly dependent: (1, –2, 1), (2, 1,…

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Q: Find the coordinates of (2, 3, 4, – 1) with respect to the basis B = {(1, 1, 1, 2), (1, –1, 0, 0),…

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Q: Let A = 5.7 ag -2.5ẩn and B = -1 âg+ 4an given the unit vectors = 0.6a+ 0.8ẩy and ẩh = 0.8ất – 0.6a…

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Q: Find the coordinate vector (x]g of x relative to the given basis B= (b1, b2, b3}. 5 b1 , b = 5 b3 =…

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Q: 6. Find a unit vector in R4 that is orthogonal to the following three vectors 1 1 α1 = , A₂= α3

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Q: 3 Find the coordinates of y 7 relative to the orthogonal basis Uj = U2 = U3 = -2 -4 10 2 2 4 252

A: The solution is given as

Q: Let ū = -4î + 4ĵ – 3k and i = î – 25 + 2k. (a) Let w = proj, ū. Compute w. (b) Compute u-w and show…

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Q: (3) Let L be the line y 2r in the plane. Find a matrix C such hat for every vector H plane, the…

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Q: Consider the vectors in M22 then S = {X₁, X2, X3} is linearly dependent. Select one: O True O False…

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Q: 5. Consider V=R². Find the coordinates of = (10, 20) with respect to the ordered basis ß= [(1, 1),…

A: we will find the coordinate vector with respect to the given basis.

Q: The vector x = H vector [2]8 = [2] A 1 B2 C3 in R² relative to the orthogonal basis B = What is the…

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Q: Find the coordinate vector [x]R of x relative to the given basis B= {b1.b2}. 3 b1 b2 - 2 X= 4 [x]g =

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Q: Q2 Given the vectors r=3i+4j-5k and s=-2J-K, where the xyz coordinate frame is obtained by rotating…

A: The given vectors are:r=3i+4j-5ks=-2j-k We have to find the cross product r×s

Q: Find the decomposition of a(t) into tangential and normal components at the point indicated. r(t) =…

A: Given a position vector rt=3-t , t+5 , t2 , t=2. To Find : Decomposition of atin terms of tangential…