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Chapter 6: Problem 3

Find the standard matrix for the linear transformation \(T\). $$T(x, y)=(2 x-3 y, x-y, y-4 x)$$

Short Answer

Expert verified
The standard matrix for the linear transformation \(T(x, y)=(2 x-3 y, x-y, y-4 x)\) is \(\[[2, -3],[1, -1],[-4, 1]]\)

Step by step solution

01

Identify the Linear Transformation

The given linear transformation \(T\) is \(T(x, y)=(2 x-3 y, x-y, y-4 x)\) represents a transformation from \(R^2\) to \(R^3\). Its standard basis vectors in \(R^2\) are \(e1 = (1,0)\) and \(e2=(0,1)\).
02

Find the Image of the Standard Basis Vectors

Apply the transformation to the standard basis vectors. Thus, \(T(e1)= T(1,0) = (2*1 - 3*0, 1*1 - 0, 0 - 4*1) = (2, 1, -4)\) and \(T(e2) = T(0,1) = (2*0 - 3*1, 0 - 1*1, 1 - 4*0) = (-3, -1, 1)\) .
03

Form the Standard Matrix

Now, the standard matrix \(A\) of the linear transformation is formed using the images of the standard basis vectors. The first column of \(A\) is \(T(e1)\) and the second column is \(T(e2)\). Hence, \(A = [[2, -3],[1, -1],[-4, 1]]\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard matrix
The concept of a standard matrix is central to understanding linear transformations. In linear algebra, the standard matrix is a representation of a linear transformation. It describes how vectors are mapped from one vector space to another.

For a given linear transformation, the standard matrix is composed of images of the standard basis vectors in one space after they have been transformed. This helps in visualizing and calculating the effect of linear transformations on any vector.
  • The columns of the standard matrix correspond to the transformed standard basis vectors.
  • This matrix allows us to apply the transformation using matrix multiplication.
  • Once the standard matrix is known, it simplifies computations required for transforming any vector in the domain space.

Understanding the standard matrix provides a direct method for applying complex transformations efficiently, especially when dealing with multi-dimensional spaces.
Basis vectors
Basis vectors are fundamental to grasping linear transformations and their matrix representations. In the context of vector spaces and transformations, basis vectors provide a minimal set of vectors that form the entire space through linear combinations.

Each vector in a vector space can be written as a combination of these basis vectors. For example, in r
  • For the transformation from R^2 to R^3, the chosen standard basis vectors in R^2 are (1, 0) and (0, 1).
  • These two vectors are sufficient as they cover all possible linear combinations in the two-dimensional space.
  • Applying a transformation to these basis vectors reveals how any vector in the space will be transformed.
By understanding basis vectors, you can see why they are the foundational building blocks of vector spaces, especially critical in creating the standard matrix.
Linear algebra
Linear algebra provides the tools to study vectors, vector spaces, and linear transformations. It is an essential area of mathematics, with applications ranging from computer graphics to systems of differential equations. At its core, linear algebra examines the properties of linear mappings between spaces.

Central to linear algebra are the concepts of vector operations and transformations. When transformations are linear, they can be expressed with matrices, making them easier to manipulate and understand.
  • Linear transformations, like the one described in the exercise, are defined by their action on vectors through matrix multiplication.
  • Matrix representation enables efficient calculation, understanding, and modification of vector transformations.
  • Key to mastering linear algebra is understanding how these transformations affect the span and dimensions of the space.
Linear algebra is a powerful tool that allows for simplified calculations and improvements in computational efficiency across various fields.

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Most popular questions from this chapter

Let \(B=P^{-1} A P .\) Prove that if \(A \mathbf{x}=\mathbf{x},\) then \(P B P^{-1} \mathbf{x}=\mathbf{x}\).

Give a geometric description of the linear transformation defined by the elementary matrix. $$A=\left[\begin{array}{rr} -1 & 0 \\ 0 & 1 \end{array}\right]$$

Determine whether the linear transformation is invertible. If it is, find its inverse. $$T(x, y)=(x+y, 3 x+3 y)$$

The linear transformation defined by a diagonal matrix with positive main diagonal elements is called a magnification. Find the images of \((1,0),(0,1),\) and (2,2) under the linear transformation \(A\) and graphically interpret your result. \(A=\left[\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right]\)

Suppose \(A\) and \(B\) are similar. Explain why they have the same rank.
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