Is matrix A in the form of RREF?Is Matrix A in RREF form?

[pyroknife]

I attached 2 problems.

For problem #1. I want to make sure I'm on the right track, to find the span of Null(A), i need to put matrix A in RREF form. By doing so I get
x1=-2t
x2=-t
x3=s
x4=u (using u because I'm using t to denote transpose)
where x1 to x4 is for each respective column. x3 and x4 turns out to be the free variables (columns 3 and 4).
This gives the vectors s[0 0 1 0]^t and u[-2 -1 0 1]^t
Thus the vectors that span Null(A) = [0 0 1 0]^t and [-2 -1 0 1]^t



The second problem is for the 2nd matrix S in the attached file. The question asks, determine whether S is a basis for the vector space V.
To do this, I think I need to take the determinant of the 3 vectors in S. This gives a determinant of "-2." This implies that the vectors are linearly indepdent and thus they form a basis for V=R^3.


Did i do these 2 problems correctly?

 

[HallsofIvy]

I attached 2 problems.

For problem #1. I want to make sure I'm on the right track, to find the span of Null(A), i need to put matrix A in RREF form. By doing so I get
x1=-2t
x2=-t
x3=s
x4=u (using u because I'm using t to denote transpose)
where x1 to x4 is for each respective column. x3 and x4 turns out to be the free variables (columns 3 and 4).

You seem to be confused as to what A is doing. A maps R³ to R⁴. The null space of A is a subspace of R³, not R⁴ and so has nothing to do with x3 and x4, which are components in R⁴. Are you clear on the definition of "null space"?

This gives the vectors s[0 0 1 0]^t and u[-2 -1 0 1]^t
Thus the vectors that span Null(A) = [0 0 1 0]^t and [-2 -1 0 1]^t

You can write a linear transformation as a matrix and put it in RREF form but it is not necessary. The definition of "null space" of A is the set of vectors that A maps to 0. So (s, t, u), in R³, will be in the null space of A if and only if -2t= 0, -t= 0, s= 0, u= 0. That's pretty clear isn't it? The null space of this a is just the 0 vector, (s, t, u)= (0, 0, 0).

The second problem is for the 2nd matrix S in the attached file. The question asks, determine whether S is a basis for the vector space V.

No, it doesn't.

To do this, I think I need to take the determinant of the 3 vectors in S. This gives a determinant of "-2." This implies that the vectors are linearly indepdent and thus they form a basis for V=R^3.


Did i do these 2 problems correctly?

For the second problem, your A is, I think,
$$\begin{bmatrix} 1 & -1 & 0 & 1 \\ 2 & -2 & 0 & 2 \\ 0 & 1 & 0 & 1\end{bmatrix}$$
which maps R⁴ to R³. Since the problem specifically says "find a set that spans the null space of A", and that is a subspace of R⁴, you need vectors in R⁴, not R³.

You seem to be intent on using "formulas" rather than thinking about what these things mean. That's a bad idea!

 

[pyroknife]

You seem to be confused as to what A is doing. A maps R³ to R⁴. The null space of A is a subspace of R³, not R⁴ and so has nothing to do with x3 and x4, which are components in R⁴. Are you clear on the definition of "null space"?

For the first problem, I was just following the example #2 on this site where the put the matrix in RREF form and found the vectors that span Null(A).

You can write a linear transformation as a matrix and put it in RREF form but it is not necessary. The definition of "null space" of A is the set of vectors that A maps to 0. So (s, t, u), in R³, will be in the null space of A if and only if -2t= 0, -t= 0, s= 0, u= 0. That's pretty clear isn't it? The null space of this a is just the 0 vector, (s, t, u)= (0, 0, 0).

But I am finding the span of Null (A).
There was a mistake in the solution I posted.
x1=-2t
x2=-t
x3=s
x4=t (I had u here previous which was wrong)

Thus,
S=[0 0 1 0]^t and [-2 -1 0 1]^t, where S spans the nullspace of A.

For the second problem, your A is, I think,
$$\begin{bmatrix} 1 & -1 & 0 & 1 \\ 2 & -2 & 0 & 2 \\ 0 & 1 & 0 & 1\end{bmatrix}$$
which maps R⁴ to R³. Since the problem specifically says "find a set that spans the null space of A", and that is a subspace of R⁴, you need vectors in R⁴, not R³.

You seem to be intent on using "formulas" rather than thinking about what these things mean. That's a bad idea!

I didn't include the problem statement for the second problem. The "find vectors that span Nullspace(A)" is for matrix A. For S, then problem statement is "determine whether S is a basis for the vector space V." This was not included in the picture I attached. In order for S to be a basis for the vector space V, the vectors inside S must be linearly independent, thus det(S)≠0. In this case the determinant is not 0,thus it forms a basis for vector space V=R^3.