Linear Combinations and Span (Concept Question)

[_N3WTON_]

Homework Statement

Let A be an m \hspace{1 mm} x \hspace{1 mm} n matrix, and let \vec{b} be a vector in \mathbb{R}^{m}. Suppose that \vec{b} is a linear combination of the columns of A. Then the columns of A span \mathbb{R}^{m}

Homework Equations

The Attempt at a Solution

I said that this statement was true using the following theorem from my textbook:
Let A be an m \hspace{1 mm}x \hspace{1 mm}n matrix. Then the following statements are logically equivalent.
a) For each \vec{b} in \mathbb{R}^{m}, the equation A \vec{x} = \vec{b} has a solution
b) Each \vec{b} in \mathbb{R}^{m} is a linear combination of the columns of A
c) The columns of A span \mathbb{R}^{m}
d) A has a pivot position in every row
However, my book says that this statement is false and I am not sure why. I think I am probably missing something obvious, but I'm not sure what.

 

[Fightfish]

b) Each \vec{b} in \mathbb{R}^{m} is a linear combination of the columns of A

let \vec{b} be a vector in \mathbb{R}^{m}. Suppose that \vec{b} is a linear combination of the columns of A.

In the statement, \vec{b} is a particular vector, and not any arbitrary vector in \mathbb{R}^{m}; that is to say, it is not necessarily true that any vector in \mathbb{R}^{m} can be expressed as a linear combination of the columns of A.

 

[_N3WTON_]

In the statement, \vec{b} is a particular vector, and not any arbitrary vector in \mathbb{R}^{m}; that is to say, it is not necessarily true that any vector in \mathbb{R}^{m} can be expressed as a linear combination of the columns of A.

ok, so I was wondering if this counter example would be a good way to verify that it is false? I picked an arbitrary matrix and an arbitrary vector:
$$
A =
\begin{bmatrix}
0 & 3\\
1& 5\\
2 &8
\end{bmatrix}
$$
$$ \vec{b} =
\begin{bmatrix}
1\\
2
\\5

\end{bmatrix} $$
I reduced A and found that there is not a pivot in every row, so I said that the columns of A do not span \mathbb{R}^{m}. Is this a sufficient counter example?

 

[Mark44]

ok, so I was wondering if this counter example would be a good way to verify that it is false? I picked an arbitrary matrix and an arbitrary vector:
$$
A =
\begin{bmatrix}
0 & 3\\
1& 5\\
2 &8
\end{bmatrix}
$$
$$ \vec{b} =
\begin{bmatrix}
1\\
2
\\5

\end{bmatrix} $$
I reduced A and found that there is not a pivot in every row, so I said that the columns of A do not span \mathbb{R}^{m}. Is this a sufficient counter example?

I'm not sure it meets the conditions of the original problem, which states that $\vec{b}$ is a linear combination of the columns of A. In any case, the condition for $\vec{b}$ seems to me to be something of a red herring. Your 3 x 2 matrix clearly (I hope) can't span R³, since there are only two columns.

 

[_N3WTON_]

I'm not sure it meets the conditions of the original problem, which states that $\vec{b}$ is a linear combination of the columns of A. In any case, the condition for $\vec{b}$ seems to me to be something of a red herring. Your 3 x 2 matrix clearly (I hope) can't span R³, since there are only two columns.

You're right that my example doesn't meet the given conditions. However, if I were to find an example that does meet the required conditions using a 3x2 matrix, could I then use that counter example to prove that the statement is false? Basically I'm still a little confused about where to go here...

 

[Mark44]

You're right that my example doesn't meet the given conditions. However, if I were to find an example that does meet the required conditions using a 3x2 matrix, could I then use that counter example to prove that the statement is false?

Yes, I believe so.

 

[_N3WTON_]

Yes, I believe so.

Awesome. I was thinking in this case it may be easier to use a matrix made up of stars and squares(like the kind used to determine echelon forms) rather than actually come up with a linear combination.

 

[Mark44]

Awesome. I was thinking in this case it may be easier to use a matrix made up of stars and squares(like the kind used to determine echelon forms) rather than actually come up with a linear combination.

No, I would use a specific matrix.

 

[_N3WTON_]

No, I would use a specific matrix.

Ok, thanks for the advice