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

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.
  • #1
kthurst
1
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"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.

Please help me out..
than u..
 
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  • #2
##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.
 

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

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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