Ranks of matrix

Question

Find the rank of the following matrix

$$\begin{bmatrix}1&-1&2\\2&1&3\end{bmatrix}$$

My approach:

The row space exists in $R^3$ and is spanned by two vectors. Since the vectors are independent of each other(because they are not scalar multiples of each other). Therefore, the row rank of the matrix which is the rank of this is two, which is the correct answer.

However, I'm still confused as to why the answer is the answer. If the row space exists in R^3 doesn't it have be be spanned by at least three vectors. For example, the unit vectors $u_1, u_2, u_3$ span row space and are independent of each other so the rank of the space should be 3.

Can someone please tell me the flaw in my logic/understanding?

Answer 1

With the case of $\mathbb{R}^3$, the dimension is 3, since it has a basis that contains 3 elements.

The row space of your matrix lives as a "subspace" of the bigger structure $\mathbb{R}^3$. That is, you don't view it as $\mathbb{R}^3$, but rather as its own entity within $\mathbb{R}^3$. It's a nicely structured chunk of $\mathbb{R}^3$, if you will.

Being its own entity, it must have its own basis! The row space is filled with linear combinations of the two rows of your matrix, and since the two rows are linearly independent (as you rightfully pointed out), its basis contains only 2 elements, so its dimension is 2!

I think an example is more enlightening. Consider this simpler matrix instead:

$$\begin{bmatrix}1&0&0\\0&1&0\end{bmatrix}$$

Its rows are linearly independent, but are elements of $\mathbb{R}^3$. In this case, your intuition wouldn't have told you that the dimension is $3$ simply because they are elements of $\mathbb{R}^3$, right? If you look at its row space (i.e., the linear combination of its rows), I'm sure you can see the plane, which is of dimension 2.

Answer 2

The definition of rank is the number of linearly independent row vectors of a matrix. For a matrix with $n$ linearly independent col, the max of rank is $n$.

Span means the linear combination of these vectors includes all vectors in this space, which means at least there are $n$ vectors(for matrix with col $n$).

Answer 3

The rank of a matrix is simply the number of nonzero rows in reduced row echelon form (rref). If you find the Reduced row echelon form of this given matrix it will yield:

(1, 0, 5/3)

(0, 1, -1/3)

Clearly, there are 2 nonzero rows in the reduced row echelon form of the given matrix. Thus, the rank is thereby 2.