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Chapter 6: Problem 71

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

Short Answer

Expert verified
In scalar multiplication, each component of a vector is multiplied by a given scalar. For example, given the vector (2,3) and scalar 3, the scalar multiplication would result in the vector (6,9).

Step by step solution

01

Definition of Scalar Multiplication

Scalar multiplication is an operation that takes a scalar (a single number) and a vector, and multiplies each component of the vector by the scalar. For any vector \( \vec{v} = v_1, v_2, ..., v_n \) and scalar \( c \), scalar multiplication is defined as \( c \cdot \vec{v} = c \cdot v_1, c \cdot v_2, ..., c \cdot v_n \).
02

Example of a Vector

For this explanation, let's take an example of a vector \( \vec{v} \) = (2,3). We'll also take a scalar \( c \) = 3.
03

Perform Scalar Multiplication

Next, multiply each component of the vector by the scalar: \( c \cdot \vec{v} \) = 3 \cdot (2,3) = (3*2, 3*3) = (6,9).
04

Verification

The final result of the scalar multiplication is the vector (6,9). Whenever you perform scalar multiplication, you should always get a vector of the same size as the original one but with its magnitude scaled by the factor of the scalar.

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Most popular questions from this chapter

Use a coding matrix \(A\) of your choice. Use a graphing utility to find the multiplicative inverse of your coding matrix. Write a cryptogram for cach message. Check your result by decoding the cryptogram. Use your graphing utility to perform all necessary matrix multiplications. $$\begin{array}{llllllllllll}\mathrm{A} & \mathrm{R} & \mathrm{T} & \- & \mathrm{E} & \mathrm{N} & \mathrm{R} & \mathrm{I} & \mathrm{C} & \mathrm{H} & \mathrm{E} & \mathrm{S} \\\1 & 18 & 20 & 0 & 5 & 14 & 18 & 9 & 3 & 8 & 5 & 19\end{array}$$

Find \(A^{-1}\) and check. $$A=\left[\begin{array}{rr}e^{x} & e^{3 x} \\\\-e^{3 x} & e^{5 x}\end{array}\right]$$

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