Why Does b Belong to R^m?: Exploring the Column Ax=b Theorem

In summary: Therefore, bεR^m because b is a matrix with 1 column and m rows, which means it belongs to R^m. So in summary, the theorem states that for an m x n matrix A, the column Ax=b has a solution for every vector b in R^m because b is a matrix with 1 column and m rows, making it belong to R^m. Additionally, in order to multiply A by vector v, v must have n components, resulting in a vector with m components. This means that A is a mapping from Rn to Rm.
  • #1
bonfire09
249
0
In this theorem it states " Let A be a m x n matrix. That is For each vector b in R^m, the column Ax=b has a solution..." Why do they say that bεR^m? Is that because b is a mx1 column matrix where it has m rows making it belong to R^m?
 
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  • #2
If A is an "m by n matrix" then it has n columns and m rows. That means that to multiply it by vector v, written as a column matrix, v must have n components, and then the product with be a vector, again written as a column matrix, will have m components. That is, A is from Rn to Rm.
 
  • #3
so what your saying is that (mxn)(mx1)=(mx1).
and that means that bεR^m?
 
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  • #4
bonfire09 said:
so what your saying is that (mxn)(mx1)=(mx1).
and that means that bεR^m?
I'm not sure I know what you mean by "(mxn)(mx1)= (mx1)". I thought we were talking about matrices, not numbers. What I said before was that a matrix with n columns and m rows, multiplied by a matrix with 1 column and n rows, gives a matrix with 1 column and n rows.

If I understand you notation, that would be "(mxn)(nx1)= (mx1)".
 
  • #5


Yes, that is correct. The notation bεR^m means that b is an element of the set of real numbers raised to the power of m, which represents the column vector with m rows. In other words, b is a vector in the m-dimensional Euclidean space, which is denoted by R^m. This is because the column vector Ax=b is defined as a linear combination of the columns of A, with the coefficients given by the entries of b. Therefore, in order for the equation Ax=b to have a solution, b must be a vector in the same space as the columns of A, which is R^m.
 

1. Why is the Column Ax=b Theorem important in linear algebra?

The Column Ax=b Theorem, also known as the Fundamental Theorem of Linear Algebra, is important because it provides a fundamental understanding of the relationships between vectors, matrices, and solutions to systems of linear equations. It serves as a basis for many other concepts and theorems in linear algebra, making it a crucial concept to understand in the field.

2. How does the Column Ax=b Theorem relate to vector spaces?

The Column Ax=b Theorem is closely related to vector spaces because it states that the set of all solutions to a system of linear equations forms a vector space. This means that the solutions can be added and scaled, and any linear combination of solutions will also be a solution. This is a fundamental property of vector spaces.

3. Can the Column Ax=b Theorem only be applied to systems of linear equations with real coefficients?

No, the Column Ax=b Theorem can be applied to systems of linear equations with any type of coefficient, including complex numbers. As long as the coefficients are in the same field, the theorem holds true.

4. How is the Column Ax=b Theorem used in practical applications?

The Column Ax=b Theorem is used in a variety of practical applications, such as in engineering, physics, and economics. It is used to solve systems of equations that arise in these fields, such as determining the forces acting on a structure, analyzing electrical circuits, or finding the optimal solution to a system of constraints.

5. Is the Column Ax=b Theorem the same as the Row Ax=b Theorem?

No, they are not the same theorem. The Column Ax=b Theorem states that the set of all solutions to a system of linear equations forms a vector space, while the Row Ax=b Theorem states that the rows of a matrix can be combined in various ways to obtain the same solution set. However, they are closely related and together provide a complete understanding of the solution set for a system of linear equations.

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