# Given the basis of find the matrix

• pyroknife
In summary, the conversation discusses the difficulty of finding a matrix given the basis for its kernel or image space. It is mentioned that knowing the kernel does not provide enough information to determine the matrix, and that the image basis is needed to determine the dimension. However, it is stated that there is no formal procedure for finding the matrix and it may require experience.
pyroknife

## Homework Statement

Not a homework problem. Typically, we are given a matrix, then asked to find the basis for the kernel or image space of the matrix. I've never seen a problem that did the converse (i.e., given the matrix for the kernel/image space of some matrix, find some matrix).

I was just wondering if anyone's got a good guideline on how to do this.

Say I have the basis for the KERNEL of matrix A formed by the vector:
##\begin{bmatrix}
0\\
1
\end{bmatrix}##.

## The Attempt at a Solution

The # of elements in the vector that forms the basis is 2, so there must be 2 columns in matrix A, but it seems matrix A can 1, 2, 3, 4...and so on # of rows. i.e.,
I can have:
1 0

or
1 0
0 0

or

1 0
0 0
0 0[/B]
.
.
.
and so on.

The basis for the kernel for any of the matrices above would have be spanned by the vector in the OP. Is this the right train of thought?

Also, I don't have a formal procedure on how to find a matrix given, the basis that forms its kernel or image space. I just obtained the previous solution by experience, i guess. Is there a formal way of thinking I should use to obtain the matrix?

pyroknife said:

## Homework Statement

Not a homework problem. Typically, we are given a matrix, then asked to find the basis for the kernel or image space of the matrix. I've never seen a problem that did the converse (i.e., given the matrix for the kernel/image space of some matrix, find some matrix).

I was just wondering if anyone's got a good guideline on how to do this.

Say I have the basis for the KERNEL of matrix A formed by the vector:
##\begin{bmatrix}
0\\
1
\end{bmatrix}##.

## The Attempt at a Solution

The # of elements in the vector that forms the basis is 2, so there must be 2 columns in matrix A, but it seems matrix A can 1, 2, 3, 4...and so on # of rows. i.e.,
I can have:
1 0

or
1 0
0 0

or

1 0
0 0
0 0[/B]
.
.
.
and so on.

The basis for the kernel for any of the matrices above would have be spanned by the vector in the OP. Is this the right train of thought?

Also, I don't have a formal procedure on how to find a matrix given, the basis that forms its kernel or image space. I just obtained the previous solution by experience, i guess. Is there a formal way of thinking I should use to obtain the matrix?

You can't do it. As you've said, knowing the kernel doesn't tell you anything about even the dimension of the image space. Knowing the image basis will tell you the dimension. But you certainly can't solve that for a specific matrix. Think about it. A lot of matrices have the same kernel and image.

Greg Bernhardt

## 1. What is a matrix?

A matrix is a mathematical object made up of numbers arranged in rows and columns. It is often used to represent data, perform calculations, and solve equations.

## 2. What is the basis of a matrix?

The basis of a matrix refers to the set of vectors that can be combined to create all of the vectors within that matrix. It is the set of fundamental vectors that make up the matrix.

## 3. How do you find the basis of a matrix?

To find the basis of a matrix, you first need to put the matrix in reduced row echelon form. Then, the columns of the matrix that contain leading 1's are the basis of the matrix.

## 4. Why is finding the basis of a matrix important?

Finding the basis of a matrix is important because it allows us to understand the structure and relationships within the data represented by the matrix. It also helps us to perform calculations and solve equations more efficiently.

## 5. Can the basis of a matrix change?

Yes, the basis of a matrix can change if the matrix is transformed through operations such as row operations or scalar multiplication. However, the basis will always contain the same fundamental vectors that make up the matrix.

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