Consider the homogeneous system Ax = 0 with A = [1 0 2 1 0; 0 1 1 0 1; 1 1 3 1 1] (rows separated by semicolons), so x has 5 unknowns. What is the dimension of the space of solutions?

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Consider the homogeneous system $A x = 0$ with $A = 101011213101011$ (rows separated by semicolons), so $x$ has $5$ unknowns. What is the dimension of the space of solutions?

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Hint

Find the rank first, then count what is left over.

Solution

Row $3$ equals row $1$ plus row $2$ , so it is redundant.
Rows $1$ and $2$ are independent, since each has a $1$ in a column where the other has a $0$ .
So the rank of $A$ is $2$ , and by the rank-nullity theorem the solution space has dimension $5 - 2 = 3$ .

Answer: $3$

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