3. Find integers p and q so that Nul A is a subspace of RP and Col A is a subspace of R' for the following matrices: [1 0 0 2 (a) 0 0 0 0 0 0 2 1 -1 -2 -5 -1 1 3 0 1 0 0 0 (b) 0 0 0 -1 -2 -1 0 0 0 ONO

Q: (2) Find a basis for the subspace W = {(a +c, 2a – b, 3a – 26 – c,b+ 2c) | a, b, c € R} of R.

A: We need to find the basis for the given subspace:

Q: Suppose V=- x = {^₁ A = [90] ₂,₂ = [a b]; C d matrixes 2 by 2 and consider S₁ = {A eV | det(A) = A…

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Q: -4 Find the orthogonal projection ŷ of y = 4 onto the subspace 3 -3 W = Span { uj U2 = 2 Ex: 1.23

A: With the help of definition of projection of a vector on a subspace, we solve this problem.

Q: 1. For each of the following either show that the given subset of R³ is a subspace of R³ or explain…

A:

Q: Which of the following must be true? OThe set of all n x matrices having trace equal to zero is a…

A:

Q: 1. For each of the following matrices, determine a basis - and N(A"): 4 subspaces R(A"), N(A), R(A),…

A: Let us apply elementary row operations on the given Matrix A to reduce it in reduced row Echelon…

Q: Which of the following is a basis for the subspace {:)- x y z W = x = y + z and x, y,z E R}C M2x3(R)…

A: We have to satisfy the given condition in the matrix i.e. x = y+z

Q: For each of the following matrices, determine a basis for each of the subspaces R(AT ), N(A), R(A),…

A:

Q: 5.7.5 ND Suppose that V has an inner product, and that n is in V. Show that the set Vo of all…

A: We will use the definition to prove that, V0 is a subspace of V, keeping in mind the properties of…

Q: 6. For each of the following subspaces W of R³, find the standard matrix of the linear trans-…

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Q: 2.14. Prove that two subspaces V1 and V2 are linearly independent if and only if Vịn V2 = {0}.

A: V1 and V2 are two subspace.

Q: Z. Suppose A and B are m x n matrices. Prove that C(A) and C(B) are orthogonal subspaces of R" if…

A:

Q: 5. (а) Show that the formula ((: :) (; 3)) = xã + 2yỹ+ zž + 2wũ gives M2x2(R) the structure of an…

A:

Q: 16. T (c)T(d) = T (c+d). 17. If the n x n matrix A is a product of elementary matrices then the null…

A: Again and again this question is coming on portal. I am answering question 17 and 18. In 16 notation…

Q: Which of the following is a basis for the subspace {:) X y z x = y+z and x, y,z €RC M2x3(R) ? 0 x 0…

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Q: Find the orthogonal complement S⊥ of the subspace S of R3 spanned by the two column vectors of the…

A:

Q: Find a basis for the following subspace of IR°. P = {¤ E R° | x1= x3 = x5, x2 + x4 = 0} What is the…

A: Explanation of the answer is as follows

Q: {(6 ) Consider the set S2 : a, b, d e R} of 2 x 2 unit upper triangular matrices. Prove this is a…

A: Introduction: A subspace is related to a vector space. It is a non-empty subset of the vector space…

Q: 2) One of the following subsets of IR2 is not subspace: a) S = {, xeR} b) s = {], = x3} %3D %3D e) s…

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Q: Find the dimension of the subspace W = {(d, c – d, c)} of R³ O Let W be the subspace of all…

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Q: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

A: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

Q: 1. Let W = {( 2).a+ 2d = 0}. Show that W is a subspace of the set of all 2 x 2 matrices and find its…

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Q: 3.8 Using the Gram-Schmidt method, turn the basis B = (b1, b2) of a two- dimensional subspace U C R…

A:

Q: 1. Which of the following sets of matrices form a subspace of M22? (assume a, b range across the…

A:

Q: Let B and C be given n x n matrices. (a) Show that U = {X € R"*" : XB = CX} and V = {YB-CY:Y € R"*"}…

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Q: Determine whether the following are subspaces of R2×2: The set of all 2 × 2 matrices A such that a12…

A:

Q: Let A and B be n × n matrices. (a) Show that AB = O if and only if the column space of B is a…

A: (a) Show that AB = O if and only if the column space of B is a subspace of the null space of A. Here…

Q: If W is a 2 x 2 matrices A with trace equal to zero is a subspace of M2,2. Prove that span X = W.…

A:

Q: Which of the following must be true? The set of all n x matrices having trace equal to zero is a…

A:

Q: 3.12. Show that for a subspace E we have (E-)- = E. Hint: It is easy to see that E is orthogonal to…

A:

Q: 2. Consider the set of matrices W : b : а,b€ R a (a) Show that W is a subspace of M22. (b) Find a…

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Q: Consider the following matrices and their SVDS -1] [1 1 0] [V2 0 0] A = U = I, E= V 1. 0 1 0] [2/V6…

A:

Q: Let A be an mxn matrix. Prove that every vector x in R" can be written in the form x = p+u, where p…

A: Let A be an m×n matrix.Prove: for evert vector x∈ℝn can be written in the form x=p+u, where p is in…

Q: Find a basis for the subspace of ps W= , X3, X4, Xs )x1 + 2x, + x5 = 0} OA {(1.0,0,1). (0,-1,0,0,2).…

A:

Q: Which of the following must be true? O The set of all n X n matrices having trace equal to zero is a…

A:

Q: A E Mrxk (F), O1 e Max (n-k) (IF), O2 e M(n-k) xk (IF) A Be M(n-k)x (n-k) (IF), | Let W S Mnxn (F).…

A:

Q: () If S is a subspace in R that is spanned by then S is spanned by: O a. 1 and 0. 1 1 and 1 c. and…

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Q: Suppose W is a set of 2 x 2 matrices A with trace A = 0 , and is a subspace of M2,2 Consider the…

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Q: Determine which of the following are subspaces of Mnn a). The set of all n × n matrices A such that…

A:

Q: let W = {x €R3 , 1 such that Ax 2 where A is any 2×3 matrix is a subspace of R³ Select one: True…

A:

Q: If A is an mxn matrix, then the row space of A is a subspace of: O A. Rmn B. R". O C. R". n-m

A: Given that A is an m×n  matrix, then the row space of A is a subspace of :

Q: 6. a. Give a basis for the orthogonal complement of the subspace V C R' given by V = {x € R* : xị…

A:

Q: An m×n matrix A is called upper triangular if all entries lying below the diagonal entries are zero,…

A: Proof:

Q: Given A be a 2 x 2 matrix with the form listed below: A = [0 a          b 0]  , where V = M2,2.…

A:

Q: For each of the following matrices, determine a basis for each of the subspaces R(AT ), N(A), R(A),…

A:

Q: Which of the following is a basis for the subspace {(: X y z W = x = y + z and x,y,z e RC M2x3(R) ?…

A:

Q: Q7) Let A be a fixed 2 x 3 matrix. Prove that the set: 1) W 2) W {x E R3: AX {x € R3: AX [1 2]'} is…

A: To prove this we have to first assume two variable that belongs to R3.

Q: Let A be an mxn matrix. Prove that W = {x ER" | Ax = 0} is a subspace of R", where W is the solution…

A:

Q: -15] 1 [2] Find the orthogonal projection of v: -11 onto the subspace V of IR spanned by and 0 -2…

A:

Q: Let W = {a + bx + cx2 + dx³| a + b = 0 and c – 3d = 0 } be a subspace of P3. Then the dimension of W…

A: