Explain why if Au vector Av vector and u vector is not equa

Explain why if Au vector = Av vector and u vector is not equal to v vector, then there is a column of A linearly dependant on the other columns.

Solution

Let us suppose that Au = Av, u v and that the columns of A are linearly independent. Then, the equation Au = 0 has only trivial solution. Then u is the zero vector. Similarly, the equation Av = 0 has only the trivial solution so that v is also a zero vector. Then u = v, which is not correct. Therefore, the columns of A are not linearly independent and therefore there is a column of A which is linearly dependant on the other columns