Nontrivial solution for Ax=0 and Ax=b determine by pivot positions [closed]

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A is a 3x2 matrix with two pivot positions.

(a) does the equation Ax=0 have a nontrivial solution

Since the two pivot positions will create 0 in the entire column in which they are present and 1 in its own position in reduced row echelon form and the rightmost column is all 0 therefore Ax=0 has no nontrivial solution

(b) does the equation Ax=b have atleast one solution for every possible b?

In the reduced row form b should have a [* * 0] form then only a unique non trivial solution exists

Is this correct and does it sound mathematical?

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1 Answer

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Your answer to (a) looks good. Question (b) can be asked alternately as $``$Can $\mathbb{R}^3$ be spanned by only two vectors in $\mathbb{R}^3$$"$?

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