3) Consider the linear transformation T such that in the standard basis its matrix is [2 1 3 A = 1 2 1 101 Take the basis B of R" given by the rows of [1 0 1] 0 1 1 1 10 a) What is the matrix B of T in the basis B? b) Compute a 3 x 3 matrix C such that B = CAC,

Q: Prove that the row vectors of an nxn invertible matrix A form a basis for R".

A: Rows of A form a basis

Q: Apply the Gram-Schmidt process to find an orthogonal basis for S. -1 4 4 1 S = = span 1 6.

A: .

Q: Let P, have the inner product (P. 4) = | P(x)q(x)dx (p, q) Apply the Gram-Schmidt process to…

A: According to the given information it is required to calculate the orthonormal basis by applying the…

Q: Find basis for the kernal and image of the linear transformation T defined by 4y + 28z 2у + 142 — 5у…

A:

Q: The Gram-Schmidt process transforms the basis u=(1, 2), U2 =(6, 2) of R into the orthogonal basis…

A:

Q: 18. Use the Gram-Schmidt process to transform the given basis (w, w, orthogonal basis where w =…

A:

Q: Find a basis for the null space of matrix A. A = 1 2 Basis is

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Q: Let Pn be the vector space of all polynomials of degree n or less in the variable x. Let D : P3 → P2…

A:

Q: The matrices A = A2 = Az A4 = form a basis for the linear space V = R²×2. Write the matrix of the…

A:

Q: (3) Let A be the matrix 1 1 -2 2 4 3 A = 1 5 -6 7 -3 1 -6 0 and let TA : Rª → Rª be the linear…

A:

Q: 3) Consider the linear transformation T such that in the standard basis its matrix is [2 1 3] A = 1…

A: The matrix is given by :  A=213121101 The standard basis of ℝ3. B=101,011,110

Q: 4. (a) Let L be the line in R' spanned by the vector (1,1, 1, 1). Find a basis for the orthogonal…

A:

Q: (c) Show that the matrices M₁ vector space M22 of 2 x 2matrices. M₂= M₁ = = 1 form a basis for the

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Prove that an m X n matrix A has rank 1 if and only if A can be written as the outer product uvT of…

A: Given that an m X n matrix A has rank 1 if and only if A can be written as the outer product uvT of…

Q: Explain how fixing a basis v1, corresponding function in n variables of the form q(x1, · ·· , Xn) =…

A:

Q: .3 Let W = span{x, x' x³, x>} in C[0,1]. Use Gram Schmidt to find an orthogonal basis for W.

A:

Q: 1. In the natural basis of vector space R³, the linear transformation p has matrix A, where (4 -8 8…

A: Given a matrix                           A=4-881-241-34 For this we compute the characteristic…

Q: Let R4 have the Euclidean inner product. Use the Gram- Schmidt process to transform the basis…

A:

Q: Show that in an n-dimensional vector space X, the representation of any x as a linear combination of…

A: The given problem is related with linear algebra. We have to show that in an n-dimensional vector…

Q: The Gram-Schimidt process transforms the basis u=(1, 3), u2-(1, -7) of R intothe orthogonal basis…

A: Given u1=⟨1,3⟩,u2=⟨1,−7⟩

Q: A vector v in an n-dimensional vector space V is said to be a cyclic vector for a linear…

A: We are given the following. Cyclic Vector: A vector v in an n-dimensional vector space V is said to…

Q: 2. Consider the quadratic form Q : R° → R, where Q(x1, 22, x3) = –2x – 8x – 18x3+ 8x1x2 + 4x1x3 +…

A: The given quadratic form is Q(x1,x2,x3)=-2x12-8x22-18x32+8x1x2+4x1x3+4x2x3 The quadratic homogenous…

Q: Let T be a linear transformation. Show that T is invertible by finding its standard matrix with…

A: NOTE: Refresh your page if you can't see any equations. . the linear transformation is so the…

Q: Let A = -1 -9 3

A:

Q: Find the matrix A of the linear transformation T(M) = om Ux2 to Ux2 (upper triangular matrices) with…

A:

Q: Recall the vector space P := {a+ bx + ca² : a, b, c E R}, with the inner product defined by (S,9) :=…

A:

Q: Suppose there is a basis a of R2, for which the system |[T]-AI|=0 has one, and only one, real…

A: Every linear operators in vector spaces with real scalars and real eigenvalues Every linear…

Q: If g and R. are bases of a vector space y and P- -1 -2 is the matrix of transition from the base B…

A:

Q: Let T be the linear transformation which rotates vectors in R2 by 60 degrees counterclockwise and…

A:

Q: Find all x in R* that are mapped into the zero vector by the transformation x-Ax for the given…

A: Answer is : X = (0 0 0 0) t

Q: If g B and B are bases of a vector space y and p= -2 is the matrix of transition from the base B to…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Consider the inner product space R, which is equipped with the stan- dard inner product. The matrix…

A: Given that B=22-2222-222. Let x,y,z∈ℝ3. Then,  φx,y,z=22-2222-222xyz=2x+2y-2z,2x+2y,2z,-2x+2y+2z And…

Q: 15. Use the Gram-Schmidt process to transform the basis {x1, 2, 23, x4} into an orthogonal basis for…

A:

Q: Determine whether the linear transfor­mation T is invertible by considering its matrix with respect…

A:

Q: 1. Consider the inner product space R?, which is equipped with the stan- dard inner product. matrix…

A:

Q: Let A be an n x n diagonalizable matrix. Suppose T : R" → R" is transformation with the standard…

A: Let A be n×n diagonalisable matrix. Suppose T:Rn→Rn be a linear transformation with the standard…

Q: (a) Find the coordinate vector of z 3 with respect to the ordered basis of R (b) Let F, be the…

A: Given that x=-5  3-1 is a coordinate vector and ordered basis is E=157,01-2,001.

Q: Let {v1, ... , vn} be an orthogonal basis for Rn and let W = span(v1, ... , vk)· Is it necessarily…

A:

Q: Find a basis B of P3, the vector space of polynomials of degree < 3, so that the transition matrix…

A: The standard basis of the vector space of polynomials, ℙ3 of degree ≤3 is,1,x,x2,x3 And the…

Q: Find the matrix for T relative to the basis B, and use Theorem 8.5.2 to compute the matrix for T…

A:

Q: The Gram-Schmidt process transforms the basis u=(1,2), u2=(2,-6) of R into the orthogonal basis…

A:

Q: The Gram-Schmidt process transforms the basis u=(1, 2), u2 =(6, 2) of R² into the orthogonal basis…

A:

Q: Show that B = {(1,1,1), (1, –1,0), (1,1, –2)} is an orthogonal basis of R³ and then expand x = (a,…

A: Let A be a matrix with the given vector as row.

Q: show that the given transformation from R2 to R2 is linear by showing that it is a matrix…

A: Given transformation from ℝ2 → ℝ2 is linear.

Q: Theorem 12. Let V be a finite dimensional inner product space and te Hom (V, V). Then linear…

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Q: 5. Consider the three linearly independent vectors uj = and let W C Rª be the span of B = {u1, u2,…

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Q: The effect of the linear transformation T : R? → R? wit with matrix A = : is to reflect each vector…

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Q: 5. a) Compute the change of basis matrix from 8 = ( ) to Ez in R?. 7() - A b) Given an explicit…

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Q: a+c\ The linear transformation T:P₂→R³ is defined by T(a+bt+ct²) = b Verify the rank nullity theorem…

A:

Q: se 3 Find a basis of the null space N(A) C R of the matrix 1. -2 A = -3 E M3x5(R) 6 -1 -7 -4 5 8 -4…

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.