# Ax=b; every vector b is exactly from one vector x (from row space of A) <more>?

• kthurst
The row space is the image, and the kernel of A is the set of vectors that are sent to zero. In summary, according to the book "Linear Algebra and its Applications" by Gilbert Starng, multiplying a matrix A with a vector x will result in every vector b being exactly from one vector x from the row space of A. This is because A defines a linear map on the vector space V and can be written as the sum of its image and kernel. This means that any vector x not in the row space will be sent to zero.
kthurst
"Ax=b; every vector b is exactly from one vector x (from row space of A)".. <more>?

Hi,
I m referring 'Linear Algebra and its applications by Gilbert Starng".

I read (ch.3.1)"Matrix transforms every vector from its row space to its column space". Or
if given Ax=b; every vector b is exactly from one vector x (from row space of A).
Just want to know What if we multiply A with some vector x which is not in row space?
(can we do it !?)...
Not able to figure it out. may be i have missed some basic concept or misunderstood it.

than u..

##A\, : \,V\longrightarrow V## defines a linear map on the vector space ##V##. We can write ##V=\operatorname{im} A \oplus \ker A##. This means we can split every vector ##v=v_{i}+ v_k## where ##v_i## is in the row space and ##v_k## is sent to zero.

## 1. What does the equation Ax=b mean in the context of vector spaces?

The equation Ax=b represents a system of linear equations, where A is a matrix and b is a vector. It means that when A is multiplied by a vector x, the resulting vector b will be a linear combination of the columns of A. In other words, b is a linear combination of the basis vectors of the column space of A.

## 2. How does the equation Ax=b relate to the row space of A?

The equation Ax=b shows that every vector b is exactly from one vector x in the row space of A. This means that the rows of A are linearly independent and span the entire vector space, allowing for a unique solution to the system of equations. The row space of A is also known as the dual space of the column space and is important in solving systems of equations.

## 3. What is the significance of every vector b being exactly from one vector x in the row space of A?

This means that the system of equations represented by Ax=b has a unique solution. In other words, there is only one vector x that satisfies the equation and produces the vector b. This is important in solving systems of equations and understanding the properties of matrices and vector spaces.

## 4. Can you provide an example of how the equation Ax=b works in practice?

Sure, let's say we have the following system of equations:
2x + 3y = 10
4x + 2y = 12
We can represent this system as Ax=b, where A is the coefficient matrix and b is the vector of constants:
A = [2 3; 4 2]
b = [10; 12]
To solve for x and y, we can use matrix operations to find x = [2; 2] and y = [2; 1]. This shows that each vector b is exactly from one vector x in the row space of A.

## 5. How is the concept of row space and the equation Ax=b used in real-world applications?

The row space of a matrix and the equation Ax=b are used in various fields such as engineering, physics, and economics. They are used to solve systems of equations, analyze data, and model real-world situations. For example, in engineering, the row space of a matrix is used to determine the forces acting on a structure, while in economics, it can be used to model supply and demand relationships. Overall, the equation Ax=b and the concept of row space are fundamental in understanding and solving problems in many different fields.

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