# Can both of this vector combine become the following matrix?

• RyozKidz
In summary, the conversation discusses the possibility of combining two vectors, A and B, into a matrix. The question is how to properly arrange the components of the matrix in relation to the original vectors. The answer lies in understanding the definition of matrix multiplication and considering how the resulting matrix should be related to the original vectors.

#### RyozKidz

vector A : ai+bj
vector B : ci+dj

can both of this vector combine become the following matrix?

$$\stackrel{a}{c}$$ $$\stackrel{b}{d}$$ or $$\stackrel{a}{b}$$ $$\stackrel{c}{d}$$

RyozKidz said:
vector A : ai+bj
vector B : ci+dj

can both of this vector combine become the following matrix?

$$\stackrel{a}{c}$$ $$\stackrel{b}{d}$$ or $$\stackrel{a}{b}$$ $$\stackrel{c}{d}$$

What do you mean by combine?

do you mean put vectors together in column or in row?

RyozKidz said:
vector A : ai+bj
vector B : ci+dj

can both of this vector combine become the following matrix?

$$\stackrel{a}{c}$$ $$\stackrel{b}{d}$$ or $$\stackrel{a}{b}$$ $$\stackrel{c}{d}$$

You just need to consider the definition of matrix multiplication. You want to find a matrix A such that

$$\left(\begin{array}{c} a\bold{i} + b\bold{j} \\ c\bold{i} + d\bold{j} \end{array}\right) = A \left(\begin{array}{c} \bold{i} \\ \bold{j} \end{array}\right) = \left(\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right) \left(\begin{array}{c} \bold{i} \\ \bold{j} \end{array}\right)$$

From this you should be able to work out the required components.

You can put numbers together any way you please and make a matrix. The question is how do you want that matrix to be related to the original vectors and what do you want to do with it?

I think i found the answer by the definition of the mutiplication of matrix...
Thx a lot..~
coz when i read about the article of the transformation matrix i can't get it..hehe thx

## 1. Can you explain what a vector is?

A vector is a mathematical object that represents a quantity with both magnitude and direction. It can be represented graphically as an arrow, with the length of the arrow indicating the magnitude and the direction of the arrow indicating the direction of the vector.

## 2. How can two vectors be combined?

Two vectors can be combined through vector addition, which involves adding the individual components of the vectors. If the vectors are in the same direction, the resulting vector will have a magnitude equal to the sum of the individual magnitudes. If the vectors are in opposite directions, the resulting vector will have a magnitude equal to the difference between the individual magnitudes. The direction of the resulting vector will depend on the direction of the individual vectors.

## 3. What is a matrix and how is it related to vectors?

A matrix is a rectangular array of numbers, symbols or expressions. It can be thought of as a way to organize and manipulate data. Matrices can be used to represent transformations of vectors, where each column of the matrix represents the transformation of a single vector. Matrices can also be used to represent systems of linear equations, where each row represents an equation.

## 4. Can any two vectors be combined to form a matrix?

No, not all combinations of vectors will result in a matrix. In order for two vectors to be combined to form a matrix, they must have the same number of components or dimensions. For example, a 2-dimensional vector and a 3-dimensional vector cannot be combined to form a matrix.

## 5. What are some real-life applications of combining vectors to form a matrix?

Combining vectors to form a matrix is a fundamental concept in linear algebra and has many real-life applications. It is used in computer graphics to represent transformations of 3D objects, in physics to represent forces acting on an object, and in machine learning for data analysis and pattern recognition. It is also used in engineering for structural analysis and optimization problems.