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Chapter 6: Problem 58

Describe how to perform scalar multiplication. Provide an example with your description.

Short Answer

Expert verified
Scalar multiplication requires each component of the vector to be individually multiplied by the scalar. For instance, the scalar multiplication of the scalar 2 and the vector \(\langle 3, 4 \rangle\) will give a result vector \(\langle 6, 8 \rangle\).

Step by step solution

01

Understanding Scalar Multiplication

From the basics of linear algebra, we know that scalar multiplication is the product of a scalar (a real number) and a vector. This can be represented as \(r \cdot \mathbf{v}\) where \(r\) is a real number (scalar) and \(\mathbf{v}\) is a vector.
02

Performing Scalar Multiplication

Before performing a scalar multiplication, ensure you know the components of your vector as well as the scalar. If we have a vector \(\mathbf{v} = \langle a, b \rangle\) and a scalar \(r\), the scalar multiplication is computed by multiplying each component of the vector by the scalar. In other words, \(r \cdot \mathbf{v} = \langle r \cdot a, r \cdot b \rangle\). Set \(r = 2\) and \(\mathbf{v} = \langle 3, 4 \rangle\) as an example.
03

Calculating the Scalar Multiplication

Now, execute the multiplication we have set up in the previous step. In this instance, our work is \(2 \cdot \langle 3, 4 \rangle = \langle 2 \cdot 3, 2 \cdot 4 \rangle\). After performing the multiplication, we find that \(2 \cdot \langle 3, 4 \rangle = \langle 6, 8 \rangle\).

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Most popular questions from this chapter

In Exercises \(1-4\) a. Give the order of each matrix. b. If \(A=\left[a_{i j}\right],\) identify \(a_{32}\) and \(a_{23}\) or explain why identification is not possible. $$ \left[\begin{array}{rrrr} -4 & 1 & 3 & -5 \\ 2 & -1 & \pi & 0 \\ 1 & 0 & -e & \frac{1}{5} \end{array}\right] $$

In Exercises \(17-26,\) let $$ A=\left[\begin{array}{rr} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{array}\right] \text { and } B=\left[\begin{array}{rr} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{array}\right] $$ Solve each matrix equation for \(X\). $$ X-A=B $$

Use Cramer's rule to solve each system or to determine that the system is inconsistent or contains dependent equations. $$ \begin{aligned}&2 x=7+3 y\\\&4 x-6 y=3\end{aligned} $$

In Exercises \(9-16,\) find: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{ll} 1 & 3 \\ 3 & 4 \\ 5 & 6 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -1 \\ 3 & -2 \\ 0 & 1 \end{array}\right] $$

Evaluate each determinant. $$ \left|\begin{array}{rrrr}-2 & -3 & 3 & 5 \\\1 & -4 & 0 & 0 \\\1 & 2 & 2 & -3 \\\2 & 0 & 1 & 1\end{array}\right| $$
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