47. The set of all skew-symmetric n x n matrices is a subspace W of Mnxn(F) (see Exercise 28 of Section 1.3). Find a basis for W. What is the dimension of W? 28. A matrix M is called skew-symmetric if Mt = -M. Clearly, a skew- symmetric matrix is square. Let F be a field. Prove that the set W1 of all skew-symmetric n x n matrices with entries from F is a subspace of Mnxn(F). Now assume that F is not of characteristic 2 (see Ap- pendix C), and let W2 be the subspace of Mnxn(F) consisting of all symmetric n x n matrices. Prove that Mnxn(F) = W1 e W2.

Q: Find a basis for the subspace of R3 spanned by S = {(2,3, -1), (1,3, -9), (0,1,5)}

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Q: Find the dimension of the subspace W of R* spanned by set S = {(3,0,1,–2), (–5,4,9,2), (–1,2,5,0)} .

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Q: 16, The set of all upper triangular n x n matrices is a subspace W of Mnxn(F) (see Exercise 12 of…

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Q: Find a basis for the subspace of R3 spanned by s. S = {(1, 5, 9), (-1, 6, 9), (2, 6, 1)}

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Q: Find a basis for the subspace of R³ spanned by S. S = {(1, 2, 6), (-1, 3, 6), (2, 3, 1)}

A: The given subspace is: S= (1, 2, 6), (-1, 3, 6), (2, 3, 1)

Q: Section 3.3: 20 Let V R7," be the vector space of real nxn matrices and let A, B E R"" n,n be such…

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Q: 12. B' = (1 + x − 2x², 3+2x + x² − 2x³,2+x+3x² − 2x³). -

A: 12. Given B'=1+x-2x2, 3+2x+x2-2x3, 2+x+3x2-2x3   Also, set S=a0+a1x+a2x2+a3x3∈ℝ3x : a0+a1+a2+3a3=0…

Q: 2: Let L be the subspace of V that contains all the real n x n lower triangular matrices. What is…

A: Given that V is the vector space and L is the subspace,which contains all real n×n lower triangular…

Q: Determine a basis and the dimension of the following subspace of R4. V = span((1, – 1,2, 4)", (1, 1,…

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Q: Let L be the subspace of V that contains all the real n x n lower triangular matrices. What is the…

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Q: Find a basis for the subspace of Rª spanned by S. S = {(29, –14, 6, –12), (-14, 7, –3, 6), (4, –2,…

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Q: Find the orthogonal projection of 5 -19 -6 onto the subspace W of R* spanned by 4 4 and 1 -2 3 projw…

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Q: Let M22 be the vector space of all real 2 × 2 matrices. Consider the subset W = {A ∈ M22 ∶ ? is an…

A: SOLUTION-  W =  { A∈ M22 : is an othogonal matrix } is not a subspace of M22,  because commutative…

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

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

A: The given set,S=2,5,-3,-2, -2,-3,2,-5, 1,3,-2,2, -1,-5,3,5

Q: (3) Suppose that V is a finite dimensional vector space over R and T € L(V) has no eigenvalues.…

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Q: 2. Provide an example of a subset U of V that is a subspace of V, as well as an example of subset W…

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Q: Find a basis for the subspace of R4 spanned by S. S = {(2, 5, -3, -4), (-2, –3, 2, -5), (1, 3, –2,…

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Q: Find a basis for the subspace of R4 spanned by S. S = {(2, 9, -2, 53), (-5, 2, 5, –2), (8, –5, –8,…

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Q: If M and N be the subspaces of V that contains the set of symmetric and skew- symmetric matrices,…

A: Given M be the subspace of symmetric matrices and N be the subspace of skew- symmetric matrices

Q: Linear algebra

A: Consider the set

Q: Find a basis for the subspace of R³ spanned by S. S = {(1, 4, 7), (-1, 5, 7), (2, 5, 1)}

A: The matrix form the vectors is: A=1-12455771

Q: Find a basis for the subspace of R3 spanned by S. S = {(5, 9, 9), (1, 2, 2), (1, 1, 1)}

A: To find the basis for the subspace of R3 spanned by the given set of vectors S.

Q: The set W of all 3 × 3 matrices of the form a subspace of M33. (See Exercise 12 in Section…

A: Consider the set W=set of all 3×3 matrices. Clearly, dimension of W is 9.

Q: [2 Is the collection of matrices a linearly independent set in M2x2? If so, does the collection…

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Q: The set of all nxn matrices A such that the linear system Ax = 0 has only trivial solutions are the…

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Q: 7. The set of all 2 x 2 upper triangular matrices is a subspace of M22- a. True. b. False.

A: Use two step test of subspace to solve this problem.

Q: Find a basis and calculate the dimension of the following subspace of R*. span{(-2,0,3,1), (1,2, –…

A: Given subspace is S=-2,0,3,1, 1,2,-1,0, -2,8,5,3, -1,2,2,1 and we have to find the basis and…

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: 2. Show that the set of all diagonal matrices of order 2 x 2 is a subspace of M2x2. That [ ] a is,…

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Q: 5. Show that the residual vector, e, can be written as e - (I, - H)y, where H is a hat matrix.…

A: As per our company guidelines we are supposed to answer only first question. Kindly repost other…

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: 4. Let Mag, denote the set of all n x n magic squares, and let Mag, denote the set of all n xn magic…

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Q: 4. Find a basis for the intersection of the subspaces V = Span ((1, 0, 1, 1), (2, 1, 1, 2)) and W =…

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Q: Find a basis of the following subspace of R“. span {(-1, 2, 1, O), (2, 0, 3, –1), (4, 4, 11, -3),…

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Q: 8) Describe geometrically all subspaces of R" for n 1, 2, 3.

A: We will do this for all three parts; n = 1, 2, and 3 (in as clear handwriting as possible)

Q: If M is linear subspace of hilbert space H then M is clos

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Q: 12) Show that subset W of a vector space V is a subspace of V if and only if span(W) = W.

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Q: Find a basis for the subspace of R4 spanned by S. S = {(3, 9, -3, 53), (-5, 3, 5, -3), (8, -5, -8,…

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Q: Is this statement true or false? EXPLAIN answer with examples, your Every linear transfor mation T:…

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Q: Find a basis for the subspace of R4 spanned by S. S = {(2, 5, -3, -4), (-2, -3, 2, -2), (1, 3, -2,…

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Q: Show whether the set W of all diagonal 3 x 3 matrices is a subspace of or

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Q: Find a basis for the subspace of R4 spanned by S.S = { (2, 9, −2, 53), (−3, 2, 3, −2), (8, −3, −8,…

A: To Find a basis for the subspace of  R4 spanned by S.…

Q: Find a basis for the subspace of R³ spanned by S. S = {(1, 5, 7), (-1, 6, 7), (2, 6, 1)}

A: Write the given set of elements as the rows  to form a 3x3 matrix. The space spanned by the rows of…

Q: Let A be an m X n matrix. Then: a. rank(ATA) = rank(A) b. The n X n matrix ATA is invertible if and…

A: As per bartleby guidelines, for more than one question, only 1st one is to be answered. Please…

Q: Find a basis for the subspace of R4 spanned by S. S = {(2, 5, -3, -5), (-2, -3, 2, -4), (1, 3, -2,…

A: We have to find the basis.

Q: Let A e Mnxn and W be the set of all solutions for the homogenous linear system AX = 0. Prove or…

A: Given, A∈Mn×n and W be the set of all solutions for the homogenous linear system AX=0. We have to…

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

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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.