Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition on S by α(x1, x2) = (αx1, αx2) (x1, x2) ⊕ (y1, y2) = (x1 + y1, 0) We use the symbol ⊕ to denote the addition operation for this system in order to avoid confusion with the usual addition x + y of row vectors. Show that S, together with the ordinary scalar multiplication and the addition operation ⊕, is not a vector space. Which of the eight axioms fail to hold?

Q: Determine k so that the vectors (1, –1, k – 1), (2, k, –4), (0, 2, + k, – 8) in R³ linearly…

A:

Q: Show that the list of vectors {r + 2, x² + 2, x³ – 3, x4 – x² + 1} is a linearly independent set of…

A:

Q: Show every vector a in R² can be written as a linear combination of and

A:

Q: Let Q be an m x n matrix with columns forming an orthonormal set. Show that (Qx) · (Qy) = x · y for…

A:

Q: (AB)† = B†A† (AB)−1 = B−1A−1

A: This is a question of linear algebra. The first one is important property regarding conjugate…

Q: 2. Prove that: k(4+ mB) - kA+(km) B for any mx n matrices A and B, and scalars k and m. (You may not…

A: 2 We have to prove that kA+mB=kA+kmB Where, A and B are m×n matrices and k and m are the scalars.…

Q: Show that if S = {v1,v2,.., Vn} is an orthogonal set of vectors, then for any scalars a1, a2, ... ,…

A: If S is an orthogonal set of vectors then to deduce that for any scalars set T is also an orthogonal…

Q: Let TA : R3 R$ be multiplication by the orthogonal matrix 1 2 - 3 3 -- 2 A = 3 2 1 Find TA(x) for…

A: Norm is square root of inner product.

Q: Let A be an invertible matrix of size 3 x 3. Let {x₁, x2. X be a set of three linearly independent…

A: Given A is an invertible matrix of size 3 x 3.  We prove that if {x1,x2,x3} be a set of three…

Q: Let x and y be vectors in Rn and let Q be an n × n orthogonal matrix. Show that if z = Qx and w = Qy…

A:

Q: Let A and B be n × n matrices and let x be a vector in Rn. How many scalar additions and…

A:

Q: Determine whether the set of vectors in P2 is linearly independent or linearly dependent.S = {1 + 3x…

A:

Q: Suppose A is a 2x2 invertible Martix. Is the following statement true or false?  For each vector b,…

A:

Q: Let W be the set of 2x1 size matrices v= with positive integer entries such that x+y is even. W is…

A: This is a problem related to matrix operations. Based on the general concept we will answer the…

Q: Write x as the sum of two vectors, one in Span (u,,u2.u3} and one in Span fu,}. Assume that {u, .ug}…

A: To find- Write x as the sum of two vectors, one in Span u1, u2, u3, and one in Span u4. Assume that…

Q: Show that the three points (x1, y1), (x2, y2), and (x3, y3) in a plane are collinear if and only if…

A:

Q: Consider four row vectors u, u, w, z of size 1 x n each and let a, bER Then. by multilinearity of…

A: The given statement is

Q: Let J, be the n x n matrix whose entries are all equal to 1, and let D(21, ..., A,) be the n X n…

A: let Jn be the n×n matrix whose entries are equal to 1 and let Dλ1,λ2,.....λn be the n×n diagonal…

Q: Show that if A is a mxn matrix and rank(A)=m then Ax=b has a solution for any vector b in Fm (Hint:…

A:

Q: Suppose that x and y are mxn matrices. Prove that xT- yT=(X -YT. Find a 2x2matrix A, that is not the…

A: Take A and B as arbitrary m×n matrices and evaluate the left and right sided expression

Q: Let A be an m×n matrix. Explain why the followingare true. Any vector x in Rn can be uniquely…

A:

Q: Let the matrices Ho, H, Ha 1x1 M어ni x [1]. be defined ... the He=L He-I vector of length n-2" can be…

A:

Q: Suppose M is the set of all real column vectors in R' whose components are equal, then the basis for…

A: See the detailed solution below.

Q: When V is spanned by {v1, v2, . . . , vk} and one of these vectors can be written as a linear…

A:

Q: Use Lagrange multipliers to find the point on the plane x – 2y+ 3z = 6 that is closest to the point…

A: Let, the point be (x,y,z) then the distance between (x,y,z) & (0,1,4) is:…

Q: Assume that th e product AB makes sense Prove that if the columns of B are linearly dependent, then…

A:

Q: Let J„ be the n × n matrix whose entries are all equal to 1, and let D(^1, ... , An) be the n × n…

A: The complete solutions are given below

Q: Suppose a is a non-zero scalar, v = (-a, 2a, a), and w = (a, -a, 2a), which of the following equals…

A: Given : v = -a, 2a, aw = a, -a, 2a

Q: Use the SVD to show that any square matrix A can be written as A = unitary and P is Hermitian. UP…

A: By SVD, we know that matrix A can be written as A=SDVTWhere S and V are orthogonal matrices andD is…

Q: Given Pauli matrices X,Y, Z ; show that X2 = Y2 = Z² = 1 (a) by using the matrix representation (b)…

A:

Q: Let A be a 7 × 5 matrix of rank 4. Let P and Q be theprojection matrices that project vectors in R7…

A:

Q: Prove as follows the inequality |Ax| =||A|| - |x|, where A is an m x m matrix with row vectors a1,…

A: Given: The order of the matrix A is m×m. The row vectors are a1,a2,a3,.....am and the dimensional…

Q: Compute the products in Exercises 1–4 using (a) the definition, as in Example 1, and (b) the…

A: Let the given matrices be A and B,A = -421601B = 3-27 Order of matrix A = 3×2Order of matrix B =…

Q: If {v1, V2, V3} is an orthogonal set of nonzero vectors in an inner product space V, then prov that…

A: This question is related to linear algebra (inner product space).

Q: Let A be an n × n matrix and let λ be a scalar. Show that det(A − λI) = 0 if and only if Ax = λx for…

A:

Q: Let A € RPXm have rank r. Show that there are matrices C and B such that A = св, where C E RPXT and…

A: Please check the solution in next step

Q: Let r1, .. r,be vectors in R", and let Q be an m xn matrix. Write the matrix Qr, Qr, as a product of…

A: Given:Let r1.......rp be vectors in ℝnLet Q be an m×n matrix Qr1.......QrpR=r1.......rpQ=Q…

Q: (a) Let M be a square matrix such that M² = 0. Let t, c be scalars and c‡0. Show that

A: We shall solve first question only as you have asked more than one different question - for others…

Q: -) Let A, B be nx n symmetric matrices such that XTAX = X™BX for all %3D column vectors X e R". Show…

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Let A be an n x n matrix. If vi, v2,..., v are vectors in R" such that {Avi, Au2,..., Avn} is…

A:

Q: Prove that if A is not square, then either the row vectors of A or the column vectors of A form a…

A: Suppose A is an x×y matrix that is not square. Either A has more rows than column (x>y) or more…

Q: Let {v1, v2} be an orthogonal set of nonzero vectors, and let c1, c2 be any nonzero scalars. Show…

A:

Q: Let u, w, v1 , U2 , and v3 be the vectors in R* defined by 9 16 19 -10 12 8. U2 = u = w = U1 = 17 U3…

A:

Q: Compute the products in Exercises 1–4 using (a) the definition, as in Example 1, and (b) the…

A: Given: A=83-4512x=111

Q: Let TA : R → R³ be multiplication by the orthogonal matrix 1 21 3 3 3 6- 1 Find TA(x) for the vector…

A:

Q: Let V be a vector space of all 4x2 matrices, then dimension of V is.. Select one: O A. 4 В. 16 с 8

A: Given that V is a vector space of all 4×2 matrices. To find the dimension of V

Q: Write x as the sum of two vectors, one in Span fu,,u2,U3} and one in Span fu4). Assume that fu. 44…

A:

Q: Show that (x, y) = 4x1y1-7x2Y2 for vectors x = (x1, x2) and y = (V1 Y2) in R2 defines an inner…

A:

Q: Let W be the set of all vectors of the form [5r – 4s – 2t] - 3r – 2s 3r – 3s + 2t 58 3r +t with r, s…

A:

Q: Let A and B be m × n matrices. (a) Suppose that v"Aw = v"Bw for all vectors v, w with real entries.…

A:

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.