**Problem 3: Linear Transformation in \( \mathbb{R}^3 \)**Consider the linear transformation \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) given by\[ T(x_1, x_2, x_3) = (x_1 + 3x_3, -2x_2 - 4x_3, 2x_1 + 6x_3). \]**Tasks:**a) Find the standard matrix for \( T \).b) Is \( T \) one-to-one? (Justify your answer.)

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Answered: 3. Consider the linear transformation…

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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With bases V1, V2, V3 andw1, w2, W3, suppose T(v1) = w2 and T(v2) = T(v3) = w1 + w3. T is a linear transformation. Find the matrix A and multiply by the vector (1, 1, 1). What is the output from T when the input is v1 + v2 + v3?
Which is correct?
2. Find the standard matrix for each linear transformation. Give some explanation. a. T:R² → R² rotates points about the origin by 180° (or ♬ radians). b. T:R² → R² reflects points about the line y = -X.
Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x,, X2, are not vectors but are entries in vectors. T(X1,X2.X3,X4) = (*1 + 9x2, 0, 4×2 + X4, X2 - X4) A = |(Type an integer or decimal for each matrix element.)
Let f be the linear transformation represented by the matrix A = 4 -2 -2-4 2). Find the factor by which f scales areas. The factor by which f scales areas is
HANAN TE The cross product of two vectors in R3 is defined by bi 6-8-6 a2b3-a3b2 a2 Xb2 azbı - a1b3 La1b2-a2b1 b3 L-₁ Let = 3. Find the matrix A of the linear transformation from R³ to R³ given by T(x) = xX. A = al a3
Let T be the linear transformation defined by T(1, 2, 3, 4, 5) = −9x1 + 7x2 + 4x3 + 6x4 + 8x5. Its associated matrix A is a matrix.
Assume that T is a linear transformation. Find the standard matrix of T. T: R² →R² first rotates points through - 3π 4 radians and then reflects points through the horizontal x₁-axis. H A = (Type an integer or simplified fraction for each matrix element. Type exact answers, using radicals as needed.)
The matrices 4₁ = [18] [18], 4₂ = [1] A3 = [8] A4 7 " = - [8], form a basis for the linear space V = R2x2. Write the matrix of the linear transformation T: R²2 R²x2 such that T(A) = 14A + 9AT relative to this basis.
Assume that T is a linear transformation. Find the standard matrix of T.
Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer. T(X1 X2 X3) = (x₁ - 4x2 +6x3, X2-9x3) (a) Is the linear transformation one-to-one? O A. T is one-to-one because the column vectors are not scalar multiples of each other. B. T is one-to-one because T(x) = 0 has only the trivial solution. O C. T is not one-to-one because the columns of the standard matrix A are linearly independent. O D. T is not one-to-one because the columns of the standard matrix A are linearly dependent.
Use the standard matrix for the linear transformation T to find the image of the vector v. T(x, y) = (x + y, x - y, 3x, 3y), v = (4, -4) T(v) =

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3. Consider the linear transformation T: R R given by T(#1, 12, T3) = (21 + 303,-22- 4x3, 201 + 6x3). a)Find the standard matrix for T. b)Is F one-to-one? (justify your answer).