In a situation where x is a nontrivial solution, then every entry in x should not be nonzero. Therefore, it's false.
<h3>What is a nontrivial solution?</h3>
A nontrivial solution simply means a system of equation where the determinant of the coefficient is zero. When x is a nontrivial solution of Ax = 0, then at least one entry should be nonzero.
The equation x= x2u + x3v, with x2 and x3 free (and neither u nor v a multiple of the other), describes a plane through the origin. This is true. Since they're free variables, the space span has dimensions and this spans through the origin.
The equation Ax = b is homogeneous if the zero vector is a solution. Since x = 0 is a solution. Therefore, Ax = b. Hence, b = 0.
The effect of adding p to a vector is to move the vector in a direction parallel to p. It should be noted that adding p to a vector moves it in a direction that's parallel.
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