Find The Values Of X X X And Y Y Y In The Following Scalar Multiplication:${ -\frac{1}{3} \cdot \begin{bmatrix} X \ -9 \end{bmatrix} = \begin{bmatrix} 2 \ Y \end{bmatrix} }$ { X = \square \} ${ Y = \square }$

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Introduction to Scalar Multiplication

Scalar multiplication is a fundamental operation in linear algebra that involves multiplying a vector by a scalar, which is a number. This operation is used to scale or stretch a vector by a certain factor. In this article, we will explore the concept of scalar multiplication and use it to find the values of $x$ and $y$ in a given equation.

Understanding the Given Equation

The given equation is a scalar multiplication of two vectors:

${ -\frac{1}{3} \cdot \begin{bmatrix} x \\ -9 \end{bmatrix} = \begin{bmatrix} 2 \\ y \end{bmatrix} }$

In this equation, the vector $\begin{bmatrix} x \\ -9 \end{bmatrix}$ is being multiplied by the scalar $-\frac{1}{3}$. The result of this multiplication is the vector $\begin{bmatrix} 2 \\ y \end{bmatrix}$.

Applying the Rules of Scalar Multiplication

To find the values of $x$ and $y$, we need to apply the rules of scalar multiplication. The rules state that when a vector is multiplied by a scalar, each component of the vector is multiplied by the scalar.

Using this rule, we can multiply the vector $\begin{bmatrix} x \\ -9 \end{bmatrix}$ by the scalar $-\frac{1}{3}$:

${ -\frac{1}{3} \cdot \begin{bmatrix} x \\ -9 \end{bmatrix} = \begin{bmatrix} -\frac{1}{3}x \\ -\frac{1}{3}(-9) \end{bmatrix} }$

Simplifying the expression, we get:

${ \begin{bmatrix} -\frac{1}{3}x \\ 3 \end{bmatrix} }$

Equating the Components

Now that we have the result of the scalar multiplication, we can equate the components of the resulting vector with the components of the given vector $\begin{bmatrix} 2 \\ y \end{bmatrix}$.

Equating the first components, we get:

${ -\frac{1}{3}x = 2 }$

To solve for $x$, we can multiply both sides of the equation by $-3$:

${ x = -3(2) }$

${ x = -6 }$

Finding the Value of $y$

Now that we have the value of $x$, we can find the value of $y$ by equating the second components of the two vectors:

${ 3 = y }$

Therefore, the value of $y$ is $3$.

Conclusion

In this article, we used the concept of scalar multiplication to find the values of $x$ and $y$ in a given equation. We applied the rules of scalar multiplication and equated the components of the resulting vector with the components of the given vector. By solving the resulting equations, we found that $x = -6$ and $y = 3$.

Final Answer

${ x = \boxed{-6} }$

${ y = \boxed{3} }$

Further Reading

For more information on scalar multiplication and linear algebra, please refer to the following resources:

• Linear Algebra by Gilbert Strang
• Scalar Multiplication of Vectors
• Linear Algebra and Its Applications by Gilbert Strang

Discussion

What are some other applications of scalar multiplication in linear algebra? How can you use scalar multiplication to solve systems of linear equations? Share your thoughts and ideas in the comments below!

Introduction

In our previous article, we explored the concept of scalar multiplication and used it to find the values of $x$ and $y$ in a given equation. In this article, we will answer some frequently asked questions about scalar multiplication and provide additional examples to help solidify your understanding of this important concept.

Q&A

Q: What is scalar multiplication?

A: Scalar multiplication is a fundamental operation in linear algebra that involves multiplying a vector by a scalar, which is a number. This operation is used to scale or stretch a vector by a certain factor.

Q: How do I apply the rules of scalar multiplication?

A: To apply the rules of scalar multiplication, you need to multiply each component of the vector by the scalar. For example, if you have a vector $\begin{bmatrix} x \\ y \end{bmatrix}$ and a scalar $c$, the result of the scalar multiplication is $\begin{bmatrix} cx \\ cy \end{bmatrix}$.

Q: Can I multiply a vector by a matrix?

A: No, you cannot multiply a vector by a matrix using scalar multiplication. However, you can multiply a vector by a matrix using matrix multiplication.

Q: How do I find the value of a variable in a scalar multiplication equation?

A: To find the value of a variable in a scalar multiplication equation, you need to isolate the variable on one side of the equation. You can do this by applying the rules of scalar multiplication and using algebraic manipulations.

Q: What are some common mistakes to avoid when working with scalar multiplication?

A: Some common mistakes to avoid when working with scalar multiplication include:

• Forgetting to multiply each component of the vector by the scalar
• Using the wrong order of operations (e.g., multiplying the scalar by the vector instead of the other way around)
• Not checking the units of the vector and the scalar to ensure that they are compatible

Q: Can I use scalar multiplication to solve systems of linear equations?

A: Yes, you can use scalar multiplication to solve systems of linear equations. By applying the rules of scalar multiplication and using algebraic manipulations, you can isolate the variables and find their values.

Examples

Example 1: Scalar Multiplication of a Vector

Find the result of multiplying the vector $\begin{bmatrix} 2 \\ 3 \end{bmatrix}$ by the scalar $4$.

Solution:

${ 4 \cdot \begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 4(2) \\ 4(3) \end{bmatrix} = \begin{bmatrix} 8 \\ 12 \end{bmatrix} }$

Example 2: Finding the Value of a Variable

Find the value of $x$ in the equation:

${ 2 \cdot \begin{bmatrix} x \\ 3 \end{bmatrix} = \begin{bmatrix} 6 \\ 6 \end{bmatrix} }$

Solution:

${ 2 \cdot \begin{bmatrix} x \\ 3 \end{bmatrix} = \begin{bmatrix} 2x \\ 6 \end{bmatrix} }$

Equating the first components, we get:

${ 2x = 6 }$

Solving for $x$, we get:

${ x = 3 }$

Conclusion

In this article, we answered some frequently asked questions about scalar multiplication and provided additional examples to help solidify your understanding of this important concept. We also discussed some common mistakes to avoid when working with scalar multiplication and showed how to use scalar multiplication to solve systems of linear equations.

Final Answer

Scalar multiplication is a fundamental operation in linear algebra that involves multiplying a vector by a scalar, which is a number. By applying the rules of scalar multiplication and using algebraic manipulations, you can find the result of scalar multiplication and solve systems of linear equations.

Further Reading

For more information on scalar multiplication and linear algebra, please refer to the following resources:

• Linear Algebra by Gilbert Strang
• Scalar Multiplication of Vectors
• Linear Algebra and Its Applications by Gilbert Strang

Discussion

What are some other applications of scalar multiplication in linear algebra? How can you use scalar multiplication to solve systems of linear equations? Share your thoughts and ideas in the comments below!