Linear Algebra - Dimension

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I Love Math

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Hello! It's me again. I am now taking Linear Algebra and I have a question about dimension.

Will there ever be a time when I am given a matrix and I'm asked to find the dimension or will I always be asked for a specific type of dimension like rank or nullity?

I ask this because I can't seem to find any exercises in my textbook where it just asks for the dimension.

My second question is that if I ever am asked for just the dimension, how would I do it?

Thanks guys!
 
Hmmm...I didn't see anything mentioning dimension on that page. I even used Ctrl-F with the word 'dimension' and didn't find anything.

Did you mean that the dimension of an mxn matrix is mxn? We learned that as the size of a matrix, not dimension.

I originally meant the dimension like rank and nullity.

I take it my instructor will have to specifically ask for the *type* of dimension.

Thank you for the link, though. I'm a big fan of MathWorld.
 
The dimension is the number of basis vectors.

For instance,

Take the matix:

\(\displaystyle \L\\A=\begin{bmatrix}1&-3&4&-2&5&4\\2&-6&9&-1&8&2\\2&-6&9&-1&9&7\\-1&3&-4&2&-5&-4\end{bmatrix}\)

By REF, the basis vectors for the row space are

\(\displaystyle \L\\\begin{bmatrix}1&-3&4&-2&5&4\end{bmatrix}\)

\(\displaystyle \L\\\begin{bmatrix}0&0&1&3&-2&-6\end{bmatrix}\)

\(\displaystyle \L\\\begin{bmatrix}0&0&0&0&1&5\end{bmatrix}\)


The column space vectors are:

\(\displaystyle \L\\\begin{bmatrix}1\\0\\0\\0\end{bmatrix}\)

\(\displaystyle \L\\\begin{bmatrix}4\\1\\0\\0\end{bmatrix}\)

\(\displaystyle \L\\\begin{bmatrix}5\\-2\\1\\0\end{bmatrix}\)


Therefore, both have dimension 3.

For that matter, the dimensions should be the same.

The dimension of the row space and column space is called the rank.

The dimension of the nullspace is the nullity.
 
I Love Math said:
Did you mean that the dimension of an mxn matrix is mxn? We learned that as the size of a matrix, not dimension.
Click to expand...
Yes, which is exactly what the definition of the dimension of a matrix is: its size or dimensions.

I Love Math said:
I originally meant the dimension like rank and nullity.
I take it my instructor will have to specifically ask for the *type* of dimension.
Click to expand...
This question is of a different sort of dimension.
This refers the dimension of a vector space, it is the cardinality of the largest set of linearly independent vectors in the space. The fact is, on a finite-dimensional vector spaces any linear transformation has a matrix representation. This is what is referred to the previous post. The rank of that matrix does give us the dimension of the image space and the null space.
 
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