EXERCICE 3: Linear maps and determinant For all aeR, let f: R² R³ be the linear map defined by fa (x, y, z) = (x + (2 (1) Write the matrix A, off in the canonical basis. (2) Compute the determinart of Ag (3) Determine the values of a such that fa is injective. e)y+z, xy +z,x - y + (4- a)z). (4) Determine the values of a such that (1,1,1) belongs to Im(fa). (5) Fora = 1, determine the kernel and the image of ₁. (6) Construct a linear map g: R² R³ such that/m g = Im fo

Q: (a) The mapping f : R² → R´ given below is a linear transformation. x1 – x2 f X2 – 1 x2 3x1 – x2 +…

A: The solution is :

Q: Q-II) 11 Use row operations to find the inverse of the natrix A 0 11 0 0

A: The given question is related with inverse of a matrix. Given that A = 111011001 . We have to find…

Q: T(x, y, z) =? (a) (x + 2y, -y- 2z, -2) (r = y, 2x = 3y + z) (b) (c) (x – y, 2x – 3y – 2) (d) - - (x…

A: Given matrix is, A=1-102-3-1 The formula for finding the linear transformation is, Tv=AvT

Q: The given T is a linear transformation from R² into R². Show that T is invertible and find a formula…

A: Given :-  The given T is a linear transformation from ℝ2 into ℝ2.  Tx1, x2 =4x1-5x2,-4x1+7x2 To…

Q: Let L: R → R be defined by L(au + Bu2) = (a+B)u1 + (a - B)u2, where u = (1, -1)', u2 = (2, 1)", and…

A:

Q: The given T is a linear transformation from R² into R². Show that T is invertible and find a formula…

A: Here, the given T is a linear transformation from ℝ2 into ℝ2. It has to be determined T is…

Q: (a) Use determinants to compute the area of the parallel- ogram spanned by the vectors (2, 3) and…

A: Note the results : Suppose we have two vectors u=<u1,u2> and v=<v1,v2> .Then area of…

Q: Let B = (v₁.....V₁) be a linearly independent set in R". Explain why B must be a basis for R".

A:

Q: Determine the following value: F20 + F30 1. 2. F34 3. F67

A:

Q: B2. (a) [2 Marks ] Show that the relation R in the set {1,2,3} given by R = {…

A:

Q: Given the coordinate matrix of x relative to a (nonstandard) basis B for R", find the coordinate…

A: Given,  The coordinate matrix of x relative to a (nonstandard) basis for Rn B = {(1,0,1), (1,1,0),…

Q: Topic is Linear Transformation: Given Function:

A:

Q: Can someone help me answer this question with all the steps thank you.

A: The given linear transformation t : 𝑅3→𝑅3 represented with respect to the standard bases by the…

Q: (C).(?)) under the standard inner product on C².

A:

Q: a) Let t: R">R be the Composite 3. linear transformation which means the vectors in R? are first…

A: Answer:

Q: Suppose that T : R² → R² is a matrix transformation and that 7([}))-[-), "([;))-}] 7((:) (a) Find…

A:

Q: a;jA = Aô;*, Show that where A is a determinant of order three and A" are cofactors of a.

A:

Q: -H 3 { and C { Let B } be two bases of R2 Find the change of coordinate matrix P_ sending a B…

A: Required to determine the change of basis matrix from B to C.

Q: The given T is a linear transformation from R² into R². Show that T is invertible and find a formula…

A:

Q: The mapping f(x, y, z) = (e* – y + z)e, + (x + e*)e2 + (e")ez satisfiesthe conditions of the inverse…

A: The given problem is to find the determinant of Jacobian matrix of the given mapping function of…

Q: Let T be a linear transformation from R³ to R² defined by T(x, y, z) = (2x + y + z, x − y − z) (*)…

A:

Q: The given T is a linear transformation from R2 into R². Show that T is invertible and find a formula…

A:

Q: The given T is a linear transformation from R² into R². Show that T is invertible and find a formula…

A: Hence T is invertible.

Q: The given T is a linear transformation from R² into R². Show that T is invertible and find a formula…

A: Introduction: The linear transformation is invertible if and only if the corresponding matrix is…

Q: Please help!

A: To establish some basic facts about the determinant of a linear transformation T from U to U (U a…

Q: Show that ? is a matrix transformation by finding its standard matrix. (solution)  (2) Find the…

A:

Q: (c) For a polynomial p = a + a₁x + a₂x² + + anx", let p denote the polynomial obtained from p by…

A:

Q: Let B = {B₁ = [], B₂ = 1²²= 1·²=6 D be a basis of M2x2 (R). Suppose that T: M₂×2 (R) → P₂(R) is a…

A:

Q: Draw the projection of b onto a and also compute it from p = îa: -=[] a = (a) b = cose 3] sin and…

A:

Q: Given the set C=(2,4,6,8}.Consider the relations R 1 and R 2 defined on C as follow: (x, y)e Rz if…

A:

Q: The given T is a linear transformation from R² into R². Show that T is invertible and find a formula…

A: