Let x and y be linearly independent vectors in Rnand let S = Span(x, y).We can use x and y to definea matrix A by settingA = xyT + yxT Show that A is symmetric.

Q: Let A be a matrixand x be a vector given as follows

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Q: b) Determine the matrix A for the linear transtormation which means that the vectors Of the room are…

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Q: Show that the column vectors of following matrix A are linerly independent

A: A matrix is a mathematical object containing numbers arranged in rows and columns. The order of a…

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

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Q: Let X be an n-dimensional random vector and the random vector Y be defined as Y = AX + b, where A…

A: It is widely used in statistics . It has applications in almost all multivariate problems .

Q: Determine all values of the parameters x and y such that the vector 1 s in the range of the above…

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Q: Find the outer product form of the SVD for the given matrix. (Enter sqrt(n) for vn.) 5 0 A = 0 5 0

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Q: Let A be a 3 x 4 matrix, let y, and y, be vectors in R3, and let w = y, + y2. Suppose y, = Ax, and…

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Q: Answer considering the matrix with the given vectors as its columns

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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…

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Q: Find all vectors in R' that are perpendicular to both [1,0, 1, 0] and [1,2,0, 1] and then find a…

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Q: 5. Give an example of a matrix A such that im(A) is in R2 is spanned by the vector A = %3D

A: Given that, A is a matrix such that imA is in ℝ2 is spanned by the vector 52

Q: For a C4 axis along z axis, please find the matrix for c, the synmetry opertion on a vector·

A: Given, C4 axis is along z axis . Upon action of C14 symmetry operation, the change with respect to x…

Q: Consider the matrix attatched and the vector u = (2 0)*. Let B=(A−AT)2 and define vector v = Bu. If…

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Q: 21 Find the matrix that projects a vector from R onto the column space of the matrix 0 12 4

A: Consider the matrix A=120124. Columns of A are 102 and 214 . Clearly, they are linearly independent.

Q: Let x and y be distinct vectors in Rn (i.e., x = y), and let A be an n × n matrix with the property…

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Q: 9. Show that the matrix 0 0 -1 0 0 1 -1 C2 = x, x- will transform the vector y into -y

A: We have to show that the matrix C2 will transform the vector in ℝ3 into the vector -x-yz. First we…

Q: Find the coordinate matrix of x = (1, 2, −1) in R3 relative to the (nonstandard) basisB′ = {u1, u2,…

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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:…

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Q: 3 2 2 1 a) X1= b) Уi%3D| - 3 y2=| 1 X2= 3 X3= 1 4 y3= 1 3 1 N O

A: X=x1,x2,x3 be set of vectors. Let A=x1,x2,x3 and AT=x1x2x3 G=AAT where G is dram matrix. If G is…

Q: Suppose T(v1) = w1 + w2 + W3 and T(v2) = w2 + w3 and T(v3) = W3. Find the matrix A for Tusing these…

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Q: Show that if annxn matrix U satisfies (Ux)· (Uy) = x•y for all x and y in Rn, then U is an…

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Q: 2. Find a nonzero vector that lies in the intersection of the plane {x + 2y + 3z = 0} 1 2 and the…

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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…

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Q: 2. If A is an m xn matrix, show that every vector in null(A) is orthogonal to every vector in…

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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: Let Bj denote the matrix obtained by replacing thejth column of the identity matrix with a vectorb =…

A: COnsider the system of equations:IX =…

Q: 4 Suppose v1, V2, V3 are vectors in R*, and the matrix A = v, v2 V3 has rank 1 W = Span{v1, V2, V3}…

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Q: Find the equilibrium vector of the matrix P : algebraically. Be sure to properly normalize the…

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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: Let  A be a symmetric tridiagonal matrix (i.e., A is symmetric and aij = 0 when ever |i−j| > 1). Let…

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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: 5. Let x and y be vectors in R" and let Q be an n x n orthogonal matrix. Show that if z = Qx and w =…

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Q: Let aj be a nonzero column vector of an m×n matrix A. Is it possible for aj to be in N(AT )?…

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Q: .. Let A = [aij] be a 3 x 3 square matrix such that aij = 1 – |i – j|- (a) Construct the matrix A…

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Q: Let A be an m x n matrix, let P be the projection matrix that projects vectors in Rm onto R(A), and…

A: The Projection matrix P is an n×n square matrix that gives a vector-space Projection from ℝn To…

Q: Let A be an m x n matrix with orthonormal columns, and x, y be in R" then prove tat the linear…

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Q: the norms of a vector X = [1,12, ...,I" and matrix A = (a5) are respectively defined as ||X|| = E…

A: Given, the norms of a vector                     X=x1, x2, . . ., xnTand matrix…

Q: Write x as the sum of two vectors, one in Span fu,,u2,u3} and one in Span (u4}. Assume that (u,.u4)…

A: Two vectors u and v is said to be orthogonal if u.v=0

Q: Suppose that A is a square matrix and that ∥Au∥ = 1 for all unit vectors u. Show that A is…

A: Given: To show that A is orthogonal, assuming that A is a square matrix. Here Au =1for all unit…

Q: +0, then the matrix H = I – 2- vv7 is orthogonal and symmetric. Show that if the vector v

A: Given  H=I-2vvTvTv

Q: Let the entries of a matrix A = \aij be defined as aij 2i2 – i+j+ g(j), where g is a %3D function of…

A: Given Ais a symmetric matrix therefore aij=aji…

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

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Q: Let A be p x p matrix such that ATA = A². Show that: (a) tr (A - AT)" (A -A") =0. (b) Show that A is…

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Q: Gve an example of a dx2 matrix A and 2-dimensional vectors u and v such that u and v are orthogonal…

A: We have to give an example of a 2×2 matrix.

Q: Let A be an 2 x 3 matrix and b be a vector in R" such that the system Ax = b has solution set 0. 0.…

A: Given: The system Ax = b has solution set 395+s30-7+t072.

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

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Q: 7. What happens if we apply the Gram-Schmidt process to a linearly dependent set of vectors? 8. Find…

A: Gram Schmidt process is applied to the set of vectors to orthonormalize them when the process is…

Q: 2. Find a 3 x 4 matrix R for which the null space of R is spanned by the vectors -- -E 0. v = u =…

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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.