2. Consider the transformation T : R4 → R³ defined byT(x1, x2, x3, x4) = (4x1 + x2 − 2x3 - 3x4, 2x1 + x2+x3 - 4x4,6x1 - 9x3+9x4)(a) Find the matrix associated with this transformation.(b) Find a basis for ker(T).(c) Find a basis for Range (T).(d) Is T one-to-one? Why?(e) Is T onto? Why?

"Since you have posted a question with multiple subparts, we will solve the first three subpartsfor…

Answered: 2. Consider the transformation T : R¹ →…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Jn is simply an nxn matrix with only ones inside.
-9 V₁ = --[-] and X₂ such that T(x) is Ax for each x. Let x = A = X₁ -2 [3] and let T: R² R² be a linear transformation that maps x into x₁v₁ +X₂V₂. Find a matrix A -8 and V₂ =
Let T and T2 two linear transformations from R? to R², defined by 7(;) - ) ()-() 3x - 2y (;) (} x+ y T1 T2 y - 3x for all (x, y)" e R?. Find the standard matrix representations for each of the following composite functions, and their inverse transformation, if they exist, by using the algorithm for matrix inversion in Lemma 3, Topic 14. )T followed by T2. (i) T2 followed by an anticlockwise rotation by angle 5 in R?. (ii) A reflection in the line y = 2x followed by a projection onto the line y = -3x in R2.
2. Let A = be an orthogonal 2 × 2 matrix. cos(0) sin(0 (a) Explain why the vector can be written as for some value of 0. a Use your answer to part (a) to find all 2×2 orthogonal matrices A = |C (b) (Hint: What can the second column of A look like?) (c) det(A) = 1? What kind of linear transformation do they correspond to? ! Given your answer to part (b), what are the 2 × 2 orthogonal matrices with
Show that T(x, y, z) = (4x + 2y – 2z, −2x + y + 3z, x - y - 2z) is not a one-to-one transformation from R³ to R³. Find a basis of the kernel of this transformation.
a) Verify the given sequence B is a basis ; and b) compute the images of the standard basis vectors with respect to the given linear transformation. 15. fi = 1+2x – 2x2, f2 = 2 + 3x – 4x2, f3 = 1 + 4x – x², B = (f1, f2, f3). - 1 T(fi) = (; 10)- 4 T(fi) = (; ).T ,T(f2) : 3 8
'
Suppose that T:R²→ R³ is a linear transformation and T(1,-3)=(4,–3, 1) and T(1,4)=(1,3, 1) and let B={(1,–3),(1,4)}. a) Find the matrix representation of Tin terms of B and S3 [T] where S3 is the standard basis for R'. b) Find the change of basis matrix [C, to convert from the standard basis in R? to B. c) Find [T] = [TCE, ->
Consider the following matrices: √/3 Q 8 M' = – 051010 100 10+ 9 (a) Find the component q in the coordinate transformation matrix Q such that the matrix is orthogonal and associated with rotation. q= = (b) Let Q transform the orthonormal basis set B to B'. Determine the components of matrix M' in the new orthonormal basis set B'. (c) Calculate the three invariants of M. I₁ I₂ = M = (d) Calculate the three invariants of M'. I I! = [2 −1 6 7 -5 6 1 -1 4 I3 = =
Let L: R3 R2 be defined by the following. X1 + X3 ***] 8x2 Suppose X1 L X₂ P = = 1 -(HH) 4 3 3 1 2 B₁ = 4 ={[8][;}} is an ordered basis for the domain and B₂ = is an ordered basis for the range. Find the matrix representation P for L relative to B₁ and B₂ such that [L(u)]b₂ = P[u]ß₁'
Suppose we have a linear transformation T where T: P3 → P₂. What are the dimensions of the matrix A for T? (a) A is a 3x3 matrix. (b) A is a 4x3 matrix. (c) A is a 3x2 matrix. (d) We can't determine the size of A without knowing the specific bases on P3 and P₂. (e) A is a 2x3 matrix.
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