Let U1, U2, U3 – be the following subspaces of R* Ui={(a, b, c, d) eR*| 2a - 3b + c + d=0, 0a - 3b - 2c - 2d=0}; U2= {a, b, c, d)eR| a=4b; c=d=0}; U3={a, b, c, d)eR| a-b+c-d=0, 2a - 2b + c + d=0}; Sub-Task 1. Find a basis and the dimension for each Ui, i=1, 2, 3 Sub-Task 2. Whether R4 is the sum of U; and Uk, Hint. You must check 3 cases (i, k)=(1, 2), (i, k)=(1, 3), (i, k)=(2, 3) Sub-Task 3. If for some pair (i, k) R4=U+Uk, check whether the sum is direct.

Q: 1. Is W=set of upper triangular 3 x 3 matrices a subspace of V=all 3 x 3 matrices? Support your…

A: I have showed that any two vectors form W linear combination of these also lies on W

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Q: Find the standard matrix for the orthogonal projection of R3 onto the subspace spanned by aj=(1,1)…

A: The vectors that span the subspace are a1=1, 1 and a2=0, 1. The normalising vectors are 12, 12 and…

Q: 2. How can you build a subspace of a vector space V by using a single vector vEV

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Q: Consider the following subspaces of R3: U=span[(2, 1, 1), (1, 2, 0)] W=span[(0, 2, 1), (0, 1, 2)]…

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Q: Let U1, U2, U3 – be the following subspaces of R4 Ui={(a, b, c, d)eR*| a - b+ 2c+ d=0, a - Ob - - .…

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Q: Find a basis for the subspace of Rº spanned by S. S = {(1, 3, 5), (-1, 4, 5), (2, 4, 1)} %3D

A: Answer

Q: Let U1, U2, Uz be the following subspaces of R: Uj = {(a,b, c) : a = c}, Show that (a) R = U + U2,…

A: Answer

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Q: Suppose that S is a subspace in R spanned by all solutions to -x,+3x,3D0. Then its orthogonal…

A: See the detailed solution below.

Q: Find a linearly independent set of vectors that spans the same subspace of R* 4 as that spanned by…

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Q: Find a basis for the subspace of R* spanned by S. S = {(43, -21, 6, 12), (-14, 7, -2, –4), (6, –3,…

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Q: 1. Find the ortho gonal puojection of the Vector (1,3,1 on the subspace of IR? Spanned by…

A: Projection

Q: Find a basis for the subspace of R4 spanned by S. S = {(-11, 6, 4, 8), (-6, 3, 2, 4), (-2, 1, 1, 2),…

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Q: Sketch all four subspaces associated to mat 1 2 A = 3 6

A: Respected student, By Bartleby guidelines we have solved all the four subspaces with the first…

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Q: Let ū be some fixed vector in R" and let S be the set of all vectors in R" that are orthogonal to ū.…

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Q: Show that [0, 1] and (0, 1] as subspaces of R with the usual topology are not homeomorphic.

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Q: Is W=set of upper triangular 3 x 3 matrices a subspace of V=all 3 x 3 matrices?

A: In this question we need to check W is a subspace. so we need to check two properties * w1 and w2…

Q: Find a basis for the subspace of R4 spanned by S. S = {(3, 9, –3, 53), (-4, 3, 4, –3), (8, –4, –8,…

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Q: Find a basis for the subspace of R4 spanned by S. S = {(3, 1, 1, 3), (-2, –1, -1, -3), (4, 2, 1, 3),…

A: Let S=3, 1, 1, 3, -2, -1, -1, -3, 4, 2, 1, 3, -12, -6, -3, -8

Q: Find the dimension of the subspace of all vectors in R° whose second and fourth entries are equal.…

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Q: Provide a situation in where it shows a union of two subspaces of a vector space V and is not…

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Q: Determine which of the following subspaces of R3 are identical: U1= span[ (1,1,-1), (2,3,-1),…

A: if the dimension of U, V and U+V are the same, then U and V both are identical. Dimension U1

Q: Is the following set of vectors in R 3 linearly dependent: {(1, 0, 3),(2, 1, -2),(0, -1, 8),(7, 2,…

A: The solution is given as

Q: Find a basis for the subspace of R* spanned by S. S = {(5, 9, –5, 53), (-3, 5, 3, -5), (8, –3, –8,…

A: A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and…

Q: 4) Find a basis for the subspace of R' spanned by the vectors. {(1,1,0,0), (0,0,1,1), (-2,0,2,2),…

A: We have fond the maximal linearly independent subset of the given set.

Q: ) Find a linearly independent set of vectors that spans the same subspace of IR" as that spanned by…

A: First we need to write given vectors in matrix form and try to use row reduced echelon form(RREF) :

Q: Is the set of noninvertible 2 × 2 matrices a subspace of R2×2 ?

A: Given, ℝ2×2=abcd: a,b,c,d∈ℝ Let M be the set of all 2×2 noninvertible matrices  i.e.…

Q: Show that [0,1] and (0,1] as subspaces of R with the usual topology are not homeomorphic.

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Q: Find a basis for the subspace of R* spanned by S. S = {(2, 5, -3, -5), (-2, -3, 2, -2), (1, 3, - 2,…

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Q: Find the projection of the vector v onto the subspace S. -1 1 -1 S = span v = -1 |-2/3 -2/3 projs v…

A: To find the projection of the vector v onto the subspace S, where

Q: Compute the projection of (1, 1, 1) onto the subspace generated by (0, 1, 0) and (-4, 1, 3)". Record…

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Q: Find a linearly independent set of vectors that spans the same subspace of R' as that spanned by the…

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Q: Find a basis for the subspace of R4 spanned by S. S = {(3, -2, -3, 6), (2, –2, –3, 6), (-1, 1, 1,…

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Q: Let v1=[-2,2,1], v2=[-8,7,4], v3=[-29,25,14], w=[6,6,1] is W in {v1,v2,v3}? how many vectors are…

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Q: 4. Let aj = (1,3, –2), a2 = (5, 1,4), and b = (6, –9, 3). (a) Project b onto the line through a1.…

A: The solution is given as follows

Q: Find a basis for the subspace of R* spanned by S. S = {(2, 5, -3, -3), (-2, –3, 2, –2), (1, 3, –2,…

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Q: Let ui = (1, –2, – 1), u2 = (2, 2, -2), and v = (0,0, 1). v is not in the subspace W spanned by u,…

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Q: Why are [0,∞) and (0,∞) which are subspaces of the real number system with the usual topology not…

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Q: determine whether the first vector is in the subspace of R3 generated by the other vectors. 1) (1,…

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Q: Find a basis for the subspace of R spanned by S. S = {(2, 5, -3,-4), (-2, -3, 2,-2), (1, 3, -2, 4),…

A: Given that S=2,5,-3,-4,-2,-3,2,-2,1,3,-2,4,-1,-5,3,2.

Q: Explain why S = {(r,y) : x, y € R, x > 0} is not a subspace of R²

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Q: Let W denote the subspace of R5 consisting of all vectors having coordinates that sum to zero. The…

A: Write the given vectors into a matrix as below and do Gauss Jordan elimination with no swapping…

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.