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Chapter 6: Problem 19

Let \(T: M_{22} \rightarrow \mathbb{R}\) be a linear transformation. Show that there are scalars \(a, b, c,\) and \(d\) such that \\[ T\left[\begin{array}{ll} w & x \\ y & z \end{array}\right]=a w+b x+c y+d z \\] for all \(\left[\begin{array}{ll}w & x \\ y & z\end{array}\right]\) in \(M_{22}\)

Short Answer

Expert verified
The transformation is given by the linear combination \(aw + bx + cy + dz\).

Step by step solution

01

Understand the Problem Setup

We need to prove that a linear transformation from the set of 2x2 matrices, denoted as \(M_{22}\), to the real numbers \(\mathbb{R}\) can be expressed in terms of the entries of a matrix. This means for any matrix \(\left[\begin{array}{ll} w & x \ y & z \end{array}\right]\) in \(M_{22}\), the transformation can be written as a linear combination of its elements.
02

Recall Properties of Linear Transformations

A linear transformation \(T: M_{22} \rightarrow \mathbb{R}\) must satisfy two main properties: additivity and homogeneity. If \(A\) and \(B\) are matrices in \(M_{22}\) and \(c\) is a scalar, then \(T(A + B) = T(A) + T(B)\) and \(T(cA) = cT(A)\).
03

Basis for \(M_{22}\)

The set \(M_{22}\) of all 2x2 matrices is a 4-dimensional vector space. A basis for this space can be considered as the matrices: \[ E_1 = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}, E_2 = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}, E_3 = \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix}, E_4 = \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} \].
04

Express \(T\) in Terms of the Basis

Given that \(T\) is linear, it can be expressed in terms of its action on the basis elements: \(T(E_1) = a\), \(T(E_2) = b\), \(T(E_3) = c\), and \(T(E_4) = d\). Therefore, for any matrix \(\left[\begin{array}{ll} w & x \ y & z \end{array}\right]\), \(T\) must satisfy: \[ T\left( wE_1 + xE_2 + yE_3 + zE_4 \right) = wT(E_1) + xT(E_2) + yT(E_3) + zT(E_4) = aw + bx + cy + dz. \]
05

Conclude the Solution

Thus, we have shown that any linear transformation \(T: M_{22} \rightarrow \mathbb{R}\) can be expressed as a linear combination of the entries of a 2x2 matrix with scalars \(a, b, c,\) and \(d\) corresponding to the transformation's action on each basis element. This completes the proof that \(T\left[\begin{array}{ll} w & x \ y & z \end{array}\right] = aw + bx + cy + dz\).

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Key Concepts

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

M_{22}
When we talk about the space denoted by \( M_{22} \), we're referring to the set of all \(2 \times 2\) matrices. This space is quite important in linear algebra because it represents a collection of matrices that have two rows and two columns.

Each matrix in \( M_{22} \) can be expressed generally in the form:
  • \(\begin{bmatrix} w & x \ y & z \end{bmatrix}\),
where \(w, x, y,\) and \(z\) are real numbers.

The concept of \( M_{22} \) is crucial when dealing with linear transformations, like in our problem, because these transformations map each matrix from \( M_{22} \) to a real number, \( \mathbb{R} \). By understanding \( M_{22} \), we can see how transformations apply to each element within these matrices.
2x2 matrix
A \(2 \times 2\) matrix is a matrix with exactly two rows and two columns. This simple structure makes it a fundamental element in matrix algebra.

Matrices of this form are used to perform basic arithmetic operations and solve linear equations more efficiently. They are small but powerful tools that can represent systems, transformations, and more in compact ways.

The four entries in a \(2 \times 2\) matrix are independent parameters that can model various situations and mathematical problems. This characteristic forms the backbone of many linear algebra problems involving transformations.
basis
The idea of a basis is foundational in linear algebra. Essentially, a basis is a set of vectors that can be combined to form any vector in a given vector space. For the vector space \( M_{22} \), which is a 4-dimensional vector space, we have a specific basis consisting of the following matrices:
  • \( E_1 = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \)
  • \( E_2 = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \)
  • \( E_3 = \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} \)
  • \( E_4 = \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} \)
These matrices are linearly independent, meaning none of them can be formed by adding multiples of the others.
By using these basis matrices, any \(2 \times 2\) matrix can be uniquely expressed as a combination of \(E_1, E_2, E_3,\) and \(E_4\).

This makes a basis highly valuable in understanding and calculating linear transformations as it simplifies how we express transformations.
linear combination
A linear combination is a mathematical operation involving multiplying matrices by scalars (real numbers) and adding the results together.

In the context of \( M_{22} \) and linear transformations, when we say that a transformation can be represented as a linear combination of a matrix's entries, we're saying that:
  • For any matrix like \(\begin{bmatrix} w & x \ y & z \end{bmatrix}\),
  • the transformation can be broken down into operations of the form \(aw + bx + cy + dz\).
Here, \(a, b, c,\) and \(d\) are constants that correspond to the transformation's impact on each basis matrix entry.

Understanding linear combinations allows us to handle and simplify complex algebraic structures, making it easier to manage transformations within vector spaces like \( M_{22} \).

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Most popular questions from this chapter

Let \(T: V \rightarrow V\) be a linear transformation such that \(T \circ T=I\) (a) Show that \(\\{\mathbf{v}, T(\mathbf{v})\\}\) is linearly dependent if and only if \(T(\mathbf{v})=\pm \mathbf{v}\) (b) Give an example of such a linear transformation with \(V=\mathbb{R}^{2}\)

Radiocarbon dating is a method used by scientists to estimate the age of ancient objects that were once living matter, such as bone, leather, wood, or paper. All of these contain carbon, a proportion of which is carbon-14, a radioactive isotope that is continuously being formed in the upper atmosphere. since living organisms take up radioactive carbon along with other carbon atoms, the ratio between the two forms remains constant. However, when an organism dies, the carbon-14 in its cells decays and is not replaced. Carbon-14 has a known half-life of 5730 years, so by measuring the concentration of carbon- 14 in an object, scientists can determine its approximate age. One of the most successful applications of radiocarbon dating has been to determine the age of the Stonehenge monument in England (Figure 6.25 ). Samples taken from the remains of wooden posts were found to have a concentration of carbon-14 that was \(45 \%\) of that found in living material. What is the estimated age of these posts?

Verify that \(S\) and \(T\) are inverses. \(S: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) defined by \(S\left\\{\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}4 x+y \\ 3 x+y\end{array}\right]\) and \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) defined by \(T\left[\begin{array}{l}x \\\ y\end{array}\right]=\left[\begin{array}{c}x-y \\ -3 x+4 y\end{array}\right]\)

Test the sets of polynomials for linear independence. For those that are linearly dependent, express one of the polynomials as a linear combination of the others. $$\left\\{x, 2 x-x^{2}, 3 x+2 x^{2}\right\\} \text { in } \mathscr{P}_{2}$$

Let \(\\{\mathbf{u}, \mathbf{v}, \mathbf{w}\\}\) be a linearly independent set of vectors in a vector space \(V\) (a) \(\mathrm{Is}\\{\mathbf{u}+\mathbf{v}, \mathbf{v}+\mathbf{w}, \mathbf{u}+\mathbf{w}\\}\) linearly independent? Either prove that it is or give a counterexample to show that it is not. (b) \(\mathrm{Is}\\{\mathbf{u}-\mathbf{v}, \mathbf{v}-\mathbf{w}, \mathbf{u}-\mathbf{w}\\}\) linearly independent? Either prove that it is or give a counterexample to show that it is not.
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