How can we know the range of linear map?

Question

If we have a linear map A, that is Ax=y, then certainly that A has a domain and range, so my question is: how can we know what the domain and the range of the linear map A are? and,

How can we find them in Maple for a particular matrix A?

Definitely if we multiply A with any vector x we will get y which will be in range of A, so what I am thinking is that, if we can find the basis of A, then multiply A with this basis we will get the basis of the range. I am not sure about that, just this is in my mind. So can anyone tell me how to find the domain and range?

Thank you.

Answer 1

I don't know Maple, so I can't help you with doing this in Maple.

However, your idea about the basis is the right one. Since every vector in the domain is a linear combination of basis vectors, multiplying a basis with $A$ certainly gives you a spanning set for the range of $A$, so the range of $A$ is the span of the images of basis vectors. However, you have no way of knowing if these vectors are linearly independent. But, by basic linear algebra, some subset of them will form a basis.

More explicitly, you can multiply $A$ by the standard basis vectors $e_i$, which are $1$ in the $i$-the coordinate and $0$ otherwise. What you get are simply the columns of $A$. Thus, the range of the linear map defined by $A$ is the span of the column vectors. You can do the usual column operations to extract a basis out of the columns of $A$.

Answer 2

If $A$ is a matrix and its determinant isn't zero, then the image is the full vector space. Otherwise you must compute the dimension of the space of vectors which get mapped to zero.

Proof: You have that $n$ linear independend vectors span the whole vector space. If the determinant of $A$ is not zero, then the column vectors, $$\{\mathbb a_1:=(a^1_i),\ \mathbb a_2:=(a^2_i),\dots\ \mathbb a_n:=(a^n_i)\}$$ are such a system. And now if you can choose the components of the vecotr $\mathbb x$ freely, then $$A\ \mathbb x=\sum_{i=1}^n\sum_{j=1}^n\ a^j_ix^i\ \mathbb e_j=\sum_{i=1}^n\left(\sum_{j=1}^n\ a^j_i\ \mathbb e_j\right) x^i=\sum_{i=1}^nx^i \mathbb a_i $$ is the span of the $\mathbb a$'s, and so any vector you choose.

Answer 3

I am assuming that the matrix of $A$, denote it by $M$, acts from the left. Then the domain will consist of all the column vectors that you can multiply by $M$ (from the left). The size of $M$ tells you how many entries those column vectors will have.

Turning our attention to the range. A general observation is that if you pick any basis of the domain and apply the linear map to them, then their images will span the range. Careful here! The images of the vectors in a basis may or may not be linearly independent, so you (more often than not) need to check that.

But when we are representing $A$ with its matrix $M$, then we can make the following observation. If we use the natural basis for the column vectors, i.e. the columns with a single $1$ in some position and $0$s elsewhere, then the images of those vectors are exactly the columns of $M$. Think about what happens to such a column vector, when you multiply it with $M$ to see that.

The conclusion is that the range is exactly the column space of $M$. Most likely Maple has a command to give you that, but I haven't used Maple for ages, so I leave that for someone else.

Answer 4

Let's say your matrix $A$ is a matrix with $m$ rows and $n$ columns over $\mathbb{R}$. Then the domain is just $\mathbb{R}^n$.

You are right, if you multiply $A$ by an element of the domain, $\mathbb{R}^n$, then you get an element of the range. If you take a basis $b_1, \dots, b_n$ of $\mathbb{R}^n$ and multiply $A$ with each of these vectors, you don't necessarily get a basis of the range, but you do get a set of vectors that span the range.

Easiest is to take the standard basis of $\mathbb{R}^n$: $e_1 = (1,0,\dots, 0)$, $e_2 = (0,1,\dots, 0)$, $\dots$, $e_n = (0, \dots, 0, 1)$. Multiplying $A$ with each of these vectors simply gives the columns of $A$.

So, the columns of $A$ span the range of $A$. To find a basis of the range, you could for instance column-reduce the matrix $A$.

In Maple, you find the column space of a matrix with the function colspace.