/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Prove that \(\left(A^{t}\right)^... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 4

Prove that \(\left(A^{t}\right)^{t}=A\) for each \(A \in \mathrm{M}_{m \times n}(F)\)

Short Answer

Expert verified
To prove that \((A^{\mathrm{T}})^{\mathrm{T}} = A\) for a given matrix \(A \in M_{m \times n}(F)\), we recall the definition of the transpose of a matrix, which exchanges the rows and columns, denoted as \(A^{\mathrm{T}}\) with elements \(a_{j i}\). We then compute the transpose of \(A^{\mathrm{T}}\), denoted as \(B\), and find its elements to be \(b_{i j} = a_{j i}\). By comparing the elements of \(A\) and \(B\), we observe that \(a_{i j} = b_{i j}\), thus proving that \((A^{\mathrm{T}})^{\mathrm{T}} = A\).

Step by step solution

01

Understand the problem

We are given a matrix A 鈭� M鈧樷倱鈧�(F) and we need to prove that (A岬�)岬� = A by working with matrix indices. Step 2: Recall the definition of transpose
02

Recall the transpose definition

The transpose of a matrix A is denoted by A岬� and is found by interchanging the rows and columns of A. If A has dimensions m x n, then A岬� has dimensions n x m. The matrix element of A is denoted as a岬⑩奔, and the corresponding matrix element of A岬� is a獗坚耽. Step 3: Compute (A岬�)岬�
03

Transpose the transpose

Now we need to find the transpose of A岬�, which we can denote as B. B has dimensions m x n since A岬� has dimensions n x m. Using the transpose definition, the matrix element of B corresponds to the matrix elements of A岬� as b岬⑩奔 = a獗坚耽. Step 4: Compare A and (A岬�)岬�
04

Compare the original matrix and the result

By comparing the matrix elements of A with the matrix elements of (A岬�)岬�, we can see that the original a岬⑩奔 elements are equal to the resulting b岬⑩奔 elements (a岬⑩奔 = b岬⑩奔). This shows that the transpose of the transpose of A is indeed equal to A itself. Step 5: Conclusion
05

End the proof

Since we have shown that the matrix elements of A and (A岬�)岬� are equal by comparing their indices (a岬⑩奔 = b岬⑩奔), we have proven that (A岬�)岬� = A for any matrix A 鈭� M鈧樷倱鈧�(F).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Linear Algebra
Linear algebra is a branch of mathematics focusing on the study of vector spaces and linear mappings between these spaces. It includes the analysis of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.

One of the fundamental operations in linear algebra is the manipulation and transformation of matrices, which are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. Matrices are used to solve systems of linear equations, to represent linear transformations, and to carry out tasks such as rotations, scaling, and translations in space. Understanding the properties and operations of matrices, such as matrix transposition, is essential for various applications in sciences, engineering, and beyond.
Transpose of a Matrix
The transpose of a matrix is one of the simplest yet most powerful operations in linear algebra. Given a matrix A, its transpose, denoted by A岬�, is achieved by flipping the matrix over its diagonal. This means that the row and column indices of each element are swapped.

For instance, if you have a 2x3 matrix, the transpose changes it into a 3x2 matrix. Formally, the element at position (i, j) in matrix A appears at position (j, i) in A岬�. In essence, the first row of A becomes the first column of A岬�, the second row of A becomes the second column of A岬�, and so on. This operation is critical in many areas because it often simplifies matrix multiplication and systems of equations, and helps compute other matrix decompositions.
Matrix Indices
In order to understand matrices and operations such as transposition, one must grasp the concept of matrix indices. Matrices are essentially composed of elements that can be referenced by their position within a grid. These positions are called indices and are usually denoted by two subscripted numbers.

The first number represents the row position and the second represents the column position. Hence, the element at the 'ith' row and 'jth' column is denoted as a岬⑩奔 for matrix A. When dealing with matrix transposition, it is crucial to keep track of these indices, as they are the key to determining the correct position of elements in the transposed matrix.
Proof Techniques
Proof techniques are essential tools in mathematics, enabling mathematicians to establish the truth of propositions and theorems. In the context of proving properties about matrices such as the transpose operation, direct proof is among the most straightforward strategies.

It involves showing that a statement holds by straightforward deduction from known facts and definitions. When dealing with matrix transposition and its properties, for example, one directly applies the definition of the transpose to show that when you transpose a matrix twice, you return to the original matrix. Other proof techniques might involve indirect proofs, proof by contradiction, or contrapositive proofs, but for many matrix properties, direct proof remains the most accessible and illuminating method.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The set of all \(n \times n\) matrices having trace equal to zero is a subspace \(W\) of \(M_{n \times n}(F)\) (see Example 4 of Section 1.3). Find a basis for W. What is the dimension of W?

Let \(\vee=\left\\{\left(a_{1}, a_{2}\right): a_{1}, a_{2} \in R\right\\}\). Define addition of elements of \(V\) coordinatewise, and for \(\left(a_{1}, a_{2}\right)\) in \(\vee\) and \(c \in R\), define $$ c\left(a_{1}, a_{2}\right)= \begin{cases}(0,0) & \text { if } c=0 \\ \left(c a_{1}, \frac{a_{2}}{c}\right) & \text { if } c \neq 0\end{cases} $$ Is \(V\) a vector space over \(R\) with these operations? Justify your answer.

Let \(S=\left\\{u_{1}, u_{2}, \ldots, u_{n}\right\\}\) be a linearly independent subset of a vector space \(\mathrm{V}\) over the field \(Z_{2}\). How many vectors are there in \(\operatorname{span}(S)\) ? Justify your answer.

Show that if $$ M_{1}=\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right), \quad M_{2}=\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right), \quad \text { and } \quad M_{3}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), $$ then the span of \(\left\\{M_{1}, M_{2}, M_{3}\right\\}\) is the set of all symmetric \(2 \times 2\) matrices.

In \(F^{n}\), let \(e_{j}\) denote the vector whose \(j\) th coordinate is 1 and whose other coordinates are 0 . Prove that \(\left\\{e_{1}, e_{2}, \ldots, e_{n}\right\\}\) is linearly independent.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.