Is Your Vector Part of the Matrix's Solution Space?

[yy205001]

Homework Statement

a)Let M be a m*n matrix and x be a n*1 coordinate vector. How can you check whether or not x is in the solution space of M?

[0 1 1 1 0]
M=[1 1 0 0 1]
[0 1 1 0 1]
[1 0 1 0 0]

b)To decide whether or not the following are in the solution space of M
i) v1=[1 0 0 1 1]^T ii) v2=[1 0 0 1 1]^T
*T means the transpose of the matrices

Any help is appreciated

Homework Equations

{x\inℝⁿ:Ax=0}

The Attempt at a Solution

I started by solving the homogeneous linear equation:
M*v1=0

[0 1 1 1 0]
[1 1 0 0 1]*[0 1 0 1 1]^T = 0
[0 1 1 0 1]
[1 0 1 0 0]

[2 2 2 0]^T ≠ 0

∴ v1 is not in the solution space of M

Am i doing the right here??

 

[HallsofIvy]

What, exactly, was the wording of the question? To ask about the "solution space" of just a matrix, M, makes no sense. We talk about the "solution space" of an equation like "Mx= b" and the answer depends on b as well as M. The solution space of "Mx= 0", with b specfically equal to 0, is the "null space" or "kernel" of matrix M.

Yes, to determine whether a given vector, x, is in the solution space of Mx= b, simply multiply M and the given x and see if the result is equal to b. Assuming that the problem is really asking whether the given x is in the "null space", so that b= 0, it is immediately clear that the top row times x does not give 0 and so x is not in the null space.

But I am still concerned about the wording of the question. If it really said "solution space", it is possible that there is some non-zero "b", perhaps given in a previous part of the problem, that you missed.