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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The standard matrix of the transformation T defined by T(x,,x2,x3)= (x,+2x2,-X3 , 4x, -X3) 1 2 0 -2 0 1 0 4 1 1 2 0 0 0 -1 40 1 1201 0 4 1 40 1 1 2 0 0 0 -1 40 -1 1 2 0 20 -1 14 -1 none
The matrix M represents a linear transformation of two-dimensional space and det(M) = 2. What can be said about the area of a region in the plane compared to the area that it is sent to under the transformation? The area of the region is one-half as large.The area of the region is one-third as large. The area of the region triples.The area of the region doubles. The linear transformation T sends all of three-dimensional space to a line. What can you say about the value of the determinant of the matrix representing the transformation? The determinant is negative.The determinant is 0. The absolute value of the determinant is less than 1.
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|
3. Consider the matrix (in Matlab notation) A = [1, a; a, 1] where the value of a gets close to 1, i.e., a = 0.9; 0.99; 0.999, etc. Study how it maps the unit circle x, y = Ax.
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.
Let the linear map f:R³ → R² and h: R² R³ M(f) be the matrix associated with f and M (h) be the matrix associated with h relative to the standard bases respectively. Then which of the following is true? " A. M(h) is a 4x2 matrix, M(f) is a 2x3 matrix and M (hof) is a 4×3 matrix. B. M (foh) is a 4×3 matrix, M (f) is a 2x3 matrix and M(h) 4x2 matrix. C. Both M (f) and M(h) are invertible since the linear maps f and h have inverses. D. M (f) is a 3x2 matrix and M(h) is 2×4 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)