# Finding a Matrix whose kernel is spanned by 2 vectors

In summary, to find a matrix whose kernel is spanned by the two given vectors u and v, one can find a vector w that is orthogonal to both u and v and create a matrix with all rows being w. This can be done by setting up a vector M(1) and using it to create a matrix M with u and v as its columns. After applying rref, the resulting matrix will be the desired matrix with an identity matrix, indicating that the kernel is spanned by u and v.

## Homework Statement

Find a matrix whose kernel is spanned by the two vectors u=(1,3,2) and v=(-2,0,4).

## The Attempt at a Solution

Tried setting vectors as a matrix and rref'ing it, but didn't know where I was getting at, also tried using an augmented identity matrix with both vectors then realized I didn't know what I was doing.

How about finding a vector w that's orthogonal to both u and v and making a matrix where all of the rows are w?

Sushil Khatri
k I think I got it...
Given:
u=[1,3,2]
v=[-2,0,4]

I put in the vector M(1)=[1,1,1]
M:=<u|v|M(1)>
and I get...
[1,-2,1
3,0,1
2,4,1]

I apply rref, and I get an identity matrix. Meaning that the above matrix is my answer, right?

That doesn't work. M*u is (-3,5,14) isn't it? If u is in the kernel M*u is supposed to (0,0,0). I don't think you heard me. Find a vector w so that u.w=0 and v.w=0. Wouldn't it work if you make a matrix with all of the rows w?

## What is a kernel and why is it important in matrix calculations?

A kernel, also known as null space, is the set of all vectors that when multiplied by a matrix result in the zero vector. It is important in matrix calculations because it represents the solutions to homogeneous linear equations, and can help determine linear independence and dimensionality of a matrix.

## Can a matrix have more than one kernel?

Yes, a matrix can have multiple kernels if it has more than one set of linearly independent vectors that span the null space.

## How do I find a matrix whose kernel is spanned by 2 given vectors?

To find a matrix with a given kernel, you can set up a system of equations using the given vectors as the basis for the kernel. Then, solve for the remaining variables in the matrix to create a matrix that satisfies the given conditions.

## What is the relationship between the kernel and the range of a matrix?

The kernel and the range of a matrix are complementary subspaces. This means that the dimension of the kernel plus the dimension of the range equals the dimension of the original vector space.

## Can a matrix have a kernel of dimension 0?

Yes, a matrix can have a kernel of dimension 0 if its columns are all linearly independent, meaning that the only solution to the equation Ax=0 is the zero vector. In this case, the matrix is invertible and its null space is only the zero vector.

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