Question

Describe all solutions to ax = 0 in parametric vector form, where a is row equivalent to the given matrix.

Answer 1

The solutions to ax = 0 are all vectors in the null space of the matrix a. These solutions can be expressed as a parametric vector form, which may involve free variables if the matrix a has any. The general solution is a linear combination of the basis vectors for this null space.

Step-by-step explanation:

The question is asking to describe all solutions to the equation ax = 0 in parametric vector form, given that a is row equivalent to a certain matrix. This matrix is understood to allow a decomposition of a vector into its perpendicular components. Such a matrix transformation typically reduces to finding the null space of the matrix.

To find the solution to ax = 0, we understand that any vector x that satisfies this equation is in the null space of a. The null vector is the most trivial solution, where all components of the vector are zero. But if there are free variables involved, the solutions may form a subspace spanned by these free variables. The general solution can then be expressed as a linear combination of these basis vectors that span the null space of a.

For instance, if our matrix a has one free variable, the parametric vector form of the solution might be t[1,0,0] if the first column of a is the pivot column and the rest are free columns. Here, t would be any scalar, representing a line through the origin in the direction of the vector [1,0,0] in three-dimensional space.

Answer 2

Describe all solutions of Ax=0
in parametric vector form, where A
is row equivalent to the given matrix.

⎡⎣⎢⎢⎢1000−20003000−610054000−610⎤⎦⎥⎥⎥

I know that I should get this into row reduced echelon form, but I'm having trouble doing so. I attempted it below.

⎡⎣⎢⎢⎢1000−20003000−610054000010⎤⎦⎥⎥⎥

⎡⎣⎢⎢⎢1000−200030000100294000010⎤⎦⎥⎥⎥

I'm not quite sure where to go from here, also I don't know how I would describe all solutions of Ax=0
.
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asked
Jan 27 '13 at 17:41

ground.clouds1
339●3●6●13 edited
Jan 13 '16 at 13:42

Martin Sleziak
40.3k●5●102●219

But it is already in echelon form. Isn't it? – Tomas Jan 27 '13 at 17:52

As a hint, try multiplying your above matrix, by a vector with six variables, say x,y,z,r,s,t
and you'll find out what x
is equal to, what r
is equal to, and what t
is equal to. – CodyBugstein Jan 27 '13 at 17:57

@Tomas For reduced echelon form, the first entry in each row must be the only entry in its column, so the original matrix isn't quite there yet. – icurays1 Jan 27 '13 at 17:58

@Tomas what? My matrix is telling me 0=1
which would make it inconsistent. – ground.clouds1 Jan 27 '13 at 17:59
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