Question

Suppose the solutions of a homogeneous system of twelve linear equations in thirteen unknowns are all multiples of one nonzero solution. Will the system necessarily have have a solution for every possible choice of constants on the right sides of the​ equations? Explain.

Answer 1

See Below

Explanation:

A non-homogeneous equation of the form Ax=b always has a solution IF:

The column space of Matrix A has equal "dimensions" and "column length".

Now,

Given, 12 equations and 13 unknown, so Matrix A can be:

Matrix A = 12 x 13

Hence, the column length = 12

Each column here is a vector in space
`R^(12)` (vector space).

Now, we essentially need to figure out if the columns span the vector space of
`R^(12)`.

The rank theorem is:

`Rank A + dim \ nulA=n`

n is 13 so,

`Rank A + dim \ nulA=13`

Hence,

Rank A = 12

Hence the dimension is 12 and columns span this.

Thus,

Thus the system always has a solution.