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Chapter 4: Problem 1

List the row vectors and column vectors of the matrix $$\left[\begin{array}{rrrr} 2 & -1 & 0 & 1 \\ 3 & 5 & 7 & -1 \\ 1 & 4 & 2 & 7 \end{array}\right]$$

Short Answer

Expert verified
The row vectors are \([2, -1, 0, 1], [3, 5, 7, -1], [1, 4, 2, 7]\), and the column vectors are \([2, 3, 1], [-1, 5, 4], [0, 7, 2], [1, -1, 7]\).

Step by step solution

01

Understand the Matrix Structure

The given matrix is:\[\begin{bmatrix}2 & -1 & 0 & 1 \3 & 5 & 7 & -1 \1 & 4 & 2 & 7 \\end{bmatrix}\] It contains 3 rows and 4 columns. Our task is to identify each row and column as individual vectors.
02

List the Row Vectors

To list the row vectors, we consider each row of the matrix as a separate vector.- Row vector 1: \(\begin{bmatrix} 2 & -1 & 0 & 1 \end{bmatrix}\)- Row vector 2: \(\begin{bmatrix} 3 & 5 & 7 & -1 \end{bmatrix}\)- Row vector 3: \(\begin{bmatrix} 1 & 4 & 2 & 7 \end{bmatrix}\)
03

List the Column Vectors

To obtain the column vectors, we take each column of the matrix individually.- Column vector 1: \(\begin{bmatrix} 2 \ 3 \ 1 \end{bmatrix}\)- Column vector 2: \(\begin{bmatrix} -1 \ 5 \ 4 \end{bmatrix}\)- Column vector 3: \(\begin{bmatrix} 0 \ 7 \ 2 \end{bmatrix}\)- Column vector 4: \(\begin{bmatrix} 1 \ -1 \ 7 \end{bmatrix}\)

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Key Concepts

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

Row Vectors
In a matrix, a row vector is essentially a horizontal slice that runs across the matrix. Each row vector contains elements located in the same horizontal line. For the matrix given in the exercise:- The first row vector is extracted from the first row, resulting in \([2, -1, 0, 1]\).- The second row vector comes from the second row, forming \([3, 5, 7, -1]\).- The third row vector is taken from the third row, which gives us \([1, 4, 2, 7]\).Each row vector has the same number of elements as there are columns in the matrix. This means that the row vectors for a 3 x 4 matrix, like the one at hand, always have four elements in a horizontal line.
Column Vectors
Column vectors are vertical slices running down a matrix. Each column vector has elements in the same vertical line across different rows of a matrix. From our sample matrix:
  • The first column vector is \([2, 3, 1]\), taken from each top position of every row.
  • The second column vector is \([-1, 5, 4]\), from the second position in every row.
  • The third column vector is \([0, 7, 2]\), focusing on the third element of each row.
  • The fourth column vector is \([1, -1, 7]\), picking the last element of every row.
Column vectors in a matrix have as many elements as there are rows. Hence, for our 3 x 4 matrix, the column vectors each contain three elements, aligning vertically one above the other.
Matrix Structure
Understanding the structure of a matrix is crucial when dealing with vectors. A matrix is essentially a rectangular arrangement of numbers, symbols, or expressions, set out in rows and columns. - The rows, which run horizontally, dictate how many row vectors exist in the matrix. In our example, there are three rows, thus three row vectors. - The columns, running vertically, determine the number of column vectors. Our matrix has four columns, resulting in four distinct column vectors. This 3 x 4 matrix signifies it is comprised of 3 rows and 4 columns. Each position within the matrix is identified by its row and column number, helping us extract individual row and column vectors easily.

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Most popular questions from this chapter

Find the standard matrix for the operator \(T\) defined by the formula. (a) \(T\left(x_{1}, x_{2}\right)=\left(2 x_{1}-x_{2}, x_{1}+x_{2}\right)\) (b) \(T\left(x_{1}, x_{2}\right)=\left(x_{1}, x_{2}\right)\) (c) \(T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}+2 x_{2}+x_{3}, x_{1}+5 x_{2}, x_{3}\right)\) (d) \(T\left(x_{1}, x_{2}, x_{3}\right)=\left(4 x_{1}, 7 x_{2},-8 x_{3}\right)\)

Show that \(T(x, y)=(0,0)\) defines a matrix operator on \(R^{2}\) but \(T(x, y)=(1,1)\) does not.

Determine whether \(T_{1} \circ T_{2}=T_{2} \circ T_{1}\) (a) \(T_{1}: R^{2} \rightarrow R^{2}\) is the orthogonal projection on the \(x\) -axis, and \(T_{2}: R^{2} \rightarrow R^{2}\) is the orthogonal projection on the \(y\) -axis. (b) \(T_{1}: R^{2} \rightarrow R^{2}\) is the rotation through an angle \(\theta_{1}\), and \(T_{2}: R^{2} \rightarrow R^{2}\) is the rotation through an angle \(\theta_{2}\) (c) \(T_{1}: R^{2} \rightarrow R^{2}\) is the orthogonal projection on the \(x\) -axis, and \(T_{2}: R^{2} \rightarrow R^{2}\) is the rotation through an angle $$\boldsymbol{\theta}$$.

Find an equation of the image of the line \(y=-4 x+3\) under multiplication by $$A=\left[\begin{array}{ll}4 & -3 \\\3 & -2\end{array}\right]$$

If \(P\) is a regular stochastic matrix with steady-state vector \(\mathbf{q},\) what can you say about the sequence of products $$P_{\mathbf{q}}, \quad P^{2} \mathbf{q}, \quad P^{3} \mathbf{q}, \ldots, \quad P^{k} \mathbf{q}, \ldots$$ $$\text { as } k \rightarrow \infty ?$$
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