**Problem 34: Linear Transformation Example**- **Objective**: Provide an example of a linear transformation whose kernel is the line spanned by the vector $\begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$ in $\mathbb{R}^3$.- **Exercise**: How would you solve this using matrix calculations such as $A \vec{x} = \vec{0}$?To find such a linear transformation, consider a matrix $A$ that when multiplied by the vector $\vec{x} = \begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$, yields the zero vector $\vec{0}$. This indicates that the vector is part of the kernel of the transformation. The process involves setting up a system of equations representing linear combinations of the rows of $A$ that result in the specified zero output for the given vector, ensuring all vectors in the kernel are scalar multiples of $\begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$.

Answered: 34. Give an example of a linear…

34. Give an example of a linear transformation whose ker- nel is the line spanned by in R³. How would you oblue this usineg Matrix caletafion such as ン

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: iv) Find a basis for ker(T). 1o lamsd zied v) Determine the dimension of the ker(T). vi) Find a…

A: Given Transformation convert to the matrix T(w)= 3-6 2-2 4 1 1-2-1-1 2 2 Transform the matrix to…

Q: 16 2 (E)- T -3 and T -15 -8 -1

Q: Let T be an linear transformation from R" to Rº. Let A be the matrix associated to T. Fill in the…

Q: Find the matrix A of the linear transformation from R to R' given by 5 T 6 x1 + -1 x2. %3D -8 7 A =

Q: Find the matrix A so that the transformation T(x) = Ar is the result of first reflecting through the…

A: We have to find the matrix A so that the transformation T(x) = Ax is the result of first reflecting…

Q: Suppose T: M2,2→P3 is a linear transformation. Let A, B, and C be the matrices given below, and…

A: Suppose T:M2,2→P3 is a linear transformation. Let A, B and C be the matrices given below, and…

Q: Prove that translation of an n-dimensional point by multiplication by an n × n matrix is not a…

Q: Suppose T : R² → R² is a linear transformation that first reflects points across the ₁-axis and then…

Q: If {~v1,··· ,~vr} is linearly independent and T is a one to one linear transformation, show that…

Q: What is the standard matrix for the linear transformation given by a clockwise rotation by 45°…

A: We will first find Reflection matrix and then rotation Later as per requirement, we will find…

Q: The matrices A₁ = [J]. - [ ] 4-[61] , 10 form a basis for the linear space V = R²x2. Write the…

Q: d 3. Find the herrel of the linear transformati L: R³ R³ with matrix 2 5 1 390 1 4 -1

A: To find kernel, we have to find null space of corresponding matrix.

Q: 4. (a) Consider the linear transformation T: R + R? that sends (E) - B- - (E) -F)- T Write down the…

Q: Determine a matrix A that describes the following linear transformation. T([*]) = [2x+³] 3y

Q: 9. Let T: Rk → RP be a Linear Transformation with standard matrix A. If T is one-to-one, which of…

A: Consider the given: If T:A→B be a linear map, then dimA≤dimBSo,T:Rk→Rp be a linear map which is one…

Q: Find the standard matrix of a linear transformation T: R³ R4 given by the formula -x₁ + x₂ + x3 3x1…

Q: Why does every linear transformation T from R 2 to R 2 take squares to parallelograms? Rectangles…

A: We have to explain whether every linear transformation T : ℝ2 → ℝ2 take squares to parallelograms,…

Q: 5. Let T: R³ R2 be the transformation defined by T (:) - = (3x - 2y + 4z 5x - 9z +42) Is T is a…

Q: 8. Let the transformation T : R' →R² be given by the formula T(x,x2,x3) = (2x, – x2, x, +3x;). Find…

Q: then the standard matrix of T is A = IfT: R² R³ is a linear transformation such that - 22 19 -5 18…

Q: 2. Please just part a, c and d

Q: If ?:?2→?2 is a linear transformation such that ?([1 6])=[25 45] and ?([2 −3])=[5 −15] then the…

Q: Let T be a linear transformation on R2 defined as follows on the standard basis of R2. T(1,0) =…

A: This question is related to linear transformations and matrix, we will solve it using given…

Q: 3. Let T: R² R² be the linear transformation which first reflects points through the line X₁ = X₂…

Q: c. Find all the points X1 x2 in R3 [x3. that are transformed to Explain.

A: Given, the transform spantial coordinates x1x2x3into plane coordinate…

Q: 10. 10a. Let T be an injective linear transformation from R to R. Let A be the matrix associated to…

Q: Use matrix multiplication to show that the linear transformation represented by A = 59 can be…

A: Given: Linear transformation represented by A=-1001. To Prove: A is the combination of reflection…

Q: Let a and B be the standard bases for R" and R" respectively. For each of the following linear…

A: By bartleby rules, only first three sub parts have been answered Given linear transformation is…

Q: 1. For the following linear transformation, T, W1 = 3x1 – 6x2 + 2x3 W2 = -2x1 + 4x2 + x3 W3 = X1 -…

A: Given: The linear transformation T, w1 = 3x1 - 6x2 + 2x3w2 = -2x1 + 4x2 + x3w3 = x1 - 2x2 - x3w4 =…

Q: Find the standard matrix for the linear transformation T (æ, y) = (2x – 3y, x – y, y – 4æ) 3y, æ –…

Q: Let S be a linear transformation from R³ to R² with associated matrix Let T be a linear…

Question

**Problem 34: Linear Transformation Example**

- **Objective**: Provide an example of a linear transformation whose kernel is the line spanned by the vector $\begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$ in $\mathbb{R}^3$.

- **Exercise**: How would you solve this using matrix calculations such as $A \vec{x} = \vec{0}$?

To find such a linear transformation, consider a matrix $A$ that when multiplied by the vector $\vec{x} = \begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$, yields the zero vector $\vec{0}$. This indicates that the vector is part of the kernel of the transformation. The process involves setting up a system of equations representing linear combinations of the rows of $A$ that result in the specified zero output for the given vector, ensuring all vectors in the kernel are scalar multiples of $\begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Linear Transformation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

? Solve
Please use Dr.Racket: How can we express Mid-Square hashing for racket program?
Construct a homogeneous transformation matrix with the provided rotation and translations. R 7 Hint: Remember that a homogeoneous matrix looks like 0 1
5. Consider a linear transformation T: R2 → R2 which reflects a vector about the line y=-x, and dilates the reflected vector by a factor of 2. Find the standard matrix for the linear transformation.
Suppose M is a linear transformation taking vectors in R² to vectors in R2, where M(1,0)=(2,-4) M(0, 1) = (-6,6) M(2,9) = (
7. Is there a linear transformation T : R³ → R³ such that T If so, what is its matrix?
5. y Let A be the linear transformation R² R2 that first reflects vectors across the line then rotates vectors 90 degrees counterclockwise abour the origin. Find the matrix representation of A. las 90⁰ >(y) [*] - [] CDC)
Suppose T: M2,2→→P2 is a linear transformation whose action on a basis for M2,2 is as follows: [18]-- Describe the action of T on a general matrix, using x as the variable for the polynomial and a, b, c, and d as constants. Use the character to indicate an exponent, e.g. ax^2-bx+c. = -2x+5 T = 0 1 1 -2 -2 = -x+4 T -4 -2 = 2x² +4x-26 T 00 = -2x²+2x+10