Problem 1Find a basis for the kernel and the image of the following maps. Also find the dimension of the kerneland image.(a) Matrix A (defined below) be a linear map A: R4 → R³(b) Let G R³ R³ be the map1 231A =1 35-23 813 -3G(x, y, z) = (x + 2y − z, y + z, x + y − 2z)

Answered: Problem 1 Find a basis for the kernel…

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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5. Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x₁, x2, and x3 are not vectors but are entries of a vector x in R³. T(x1, x2, x3) = (2x1 — x2, x1 − 2x2 + x3, x1 − 3x2 + x3) - X1
Suppose B = {b1, b2} is a basis for V. C = {C₁, C₂, C3 is a basis for W. Let T: V→ W be a linear transformation with the property that: T (b₁) = 3c₁ - 2₂ + 5c3 y T (b₂) = 4¯₁ + 7C₂-C3 Find the matrix M for T relative to B and C. Determine the kernel and image of T.
7. Let T: R³ R³ be defined by (a) (b) (E) x1 + x2 + x3 = X1 X2 X3 [X1X2 X3] Find the matrix of linear transformation for T. T Find all vectors 7 € R³ such that T(x) = x.
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
Suppose T is a linear transformation on R² (i.e. T(x, y) = (ax + by, cx + dy) where a, b, c, d are real numbers and, interpreting T as a matrix, det(T) # 0. Let P be the parallelogram which is the image of the square D = = [0, 1] × [0, 1] (i.e. T(D) =P). Then the area of P is given by O lad - bcl ad bc abcd 1 |ad - bc| O labcd|
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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Problem 1 Find a basis for the kernel and the image of the following maps. Also find the dimension of the kernel and image. (a) Matrix A (defined below) be a linear map A: R4 → R³ (b) Let G R³ R³ be the map 1 2 3 1 A = 1 3 5 -2 3 8 13 -3 G(x, y, z) = (x + 2y − z, y + z, x + y − 2z)