In the vector space R2 consider the two bases given by [u1, u2] with u1 = (1, 2)T, u2 = (2, 5)T and [v1, v2] with v1 = (3, 2)T, v2 = (4, 3)T (both with respect to the standard basis). Follow these steps to find transition matrices from [v1, v2] to [u1, u2] and back again: a) Find the transition matrix V from [v1, v2] to the standard basis [e1, e2]. b) Find the transition matrix U from [u1, u2] to the standard basis [e1, e2]. c) Find the transition matrix from [v1, v2] to [u1, u2].

Q: Find the 2 x 2 matrix P such that Po is the orthogonal projection of in R2 onto the line consisting…

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Q: Prove that the vectors (1,1,0), (3,1,3) and (5,3,3) are linearly dependent.

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Q: Find the coordinate matrix of w = (5, −5, 2) relative to the orthonormal basis B for R3 below.

A: Basis: A set of vectors S is said to be a basis of the vector space V if the set S is linearly…

Q: A)  Justify if the matrix A is diagonalizable.B)  Find a matrix T and a diagonal matrix J, such that…

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Q: The vectors --[+]»-[i]»-[3]m² -]»-[3]»=[] V2 = span R2 but do not form a basis. Find two different…

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Q: The dot product of two vectors in R3 is defined by = a¡b1 + azb2 + azb3. [b3] a2 b2 8 Let i Find the…

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Q: Show linear dependence in part (ii) to show that ({1,0,1}) is a basis.

A: From part ii)We have to show that the set  is a basis of .We know thatA basis B of a vector space V…

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

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Q: at vectors （c1 - a1, c2 - a2）and （b1-a1,b2-a2）are linearly depend

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Q: Show that the column vectors (1,1,2,4), (2,-1,-5,2), (1,-1,-4,0) and (2,1,1,6) vectors in R are…

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Q: → Suppose TR³ R³ be a linear bronsformation whose standard matrix has the basis (o 1, 2), (2, 1…

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Q: Denote the column vectors of the matrix 11,922.. , ann +] True, false, or uncertain: a; is…

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Q: The second column of the matrix of change basis from the basis B={u= (1, 1, 1); v = (1,0,1); w=…

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Q: The second column of the matrix of change basis from the basis B = {u= (1,1,1); v = (1,0,1); w=…

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Q: 6 Let b₁=[2] b₂ = 3 4 [ C₁ = [ ² ] ² ₂ = 1²²] > V = [] (2 13 2 and C = { C₁₂ (₂} Find and consider…

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Q: d. Find the transition matrix from C to B. e. Find the coordinates of u = (2, –2) in the ordered…

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Q: Determine whether the set of vectors {(2,-1,3),(4,1,2),(8,-1,8)} form a basis for R3.

A: Consider the vectors : 2,-1,3, 4,1,2, 8,-1,8 To form a basis these vectors, the vectors must be…

Q: Decide whether the vectors [2, 9, -1, 4, 7, 8, 10, 9], [11, 2, 5, -9, 0, 0, -1, 3], [4, 18, -2, 8,…

A: We have to decide whether the given vectors are linearly independent. Then, we can proceed further.

Q: (i) Prove that { (1, 2, 1) , (2, 1, 5) , (1, –4, 7) } is linear dependent subset of R³. (ii)…

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Q: Determine whether the vectors (1,2,3), (5,6,7), (9,10,11), and (4,8, 12) are linearly independent.

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Q: In P2, find the the standard basis C = {1,t,t²}. Then find the B-coordinate vector for 1 - 2t + 2t².…

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Q: 5. Determine whether vectors u =(2,-6,-5), v = (3,8, 18) and w = (8,-6,7) will form a basis for R³.…

A: Basis is very important in linear algebra. A basis set always contains linearly independent vectors…

Q: Find orthogonal bases for the null space and for the range of the matrix  [1, 0, 0, 3; 1, 1, 0, 4;…

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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: b) Express the vector (–1,5, -6) in R³ as a linear combination of (2,0, 7), (2,4, 5) and(2, –12,13).

A: See the detailed solution below.

Q: Prove that the vectors (1,1,0),(3,1,3) and (5,3,3)  are linearly dependent

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Q: Q.2) For VR2 and B = {(1,2), (3, 4)"}, C = {(7,3), (4,2)"}; consider one vector v = = (1,0) ¹. •…

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Q: Construct an orthonormal set of vectors in ℝ3 from the set of linearly independent vectors ? =…

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Q: Make the vectors u1=(2,0,-1,1), u2=(1,1,-1,3), u3= (-1,1,1,-1) orthonormal.

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Q: Let the basis B = {b1, b2} and basis C = {c1, c2} where vectors b1 = (7 -2). b2 = 2 -1 , c1 = 4 1…

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Q: Find a non-zero vector in Nul A and a non-zero vector in Col A for the following matrix: 1 2 3 4 5 7…

A: We will first find a basis for Nul A and Col A then we will find the required vectors.

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.