Proving the Existence of a Solution for Ax = b in Rn: The Importance of Spanning the Range of A

If A is an n x n matrix, then the equation Ax = b has at least one solution for each b in Rn.

To show that the equation Ax = b has at least one solution for each b in Rn, we need to prove that the range of A spans the entire Rn space

To show that the equation Ax = b has at least one solution for each b in Rn, we need to prove that the range of A spans the entire Rn space.

Let’s assume that A is an n x n matrix and b is a vector in Rn.

By definition, the range of A is the set of all possible outputs that can be obtained by multiplying A with a vector. We can represent this set as Range(A).

To prove that the Range(A) spans the entire Rn space, we need to show that for any vector b in Rn, there exists a vector x such that Ax = b.

We can express the equation Ax = b as a linear combination of the columns of A:

A * [x1, x2, …, xn] = [b1, b2, …, bn],

where x1, x2, …, xn are the unknown entries of vector x, and b1, b2, …, bn are the entries of vector b.

This equation is equivalent to:

[x1 * column1 + x2 * column2 + … + xn * columnn] = [b1, b2, …, bn],

where column1, column2, …, columnn are the columns of matrix A.

To find a solution for this equation, we need to find values for x1, x2, …, xn such that the left-hand side of the equation equals the right-hand side.

Since we are showing that the range of A spans the entire Rn space, it means that any vector b in Rn can be obtained by a linear combination of the columns of A. In other words, any b can be written as a linear combination of the vectors column1, column2, …, columnn.

Therefore, the equation Ax = b has at least one solution for each b in Rn, because the columns of A span the entire Rn space, and we can find suitable values for x such that the linear combination of the columns of A equals the given vector b.

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