# Finding Kernel, Image, Rank, Nullity of Matrix

• Marakh
In summary, the kernel of the given matrix is the set of all vectors that satisfy the equations a- b+ c+ d= 0, a+ 2b- c+ d= 0, and 3b- 2c= 0. The image is the set of all vectors that can be written as Ax, where x is a vector with components (a, b, c, d). The rank of the matrix is 2 and the nullity is 2.
Marakh

## Homework Statement

Find Kernel, Image, Rank and Nullity of the matrix
1 −1 1 1 
| 1 2 −1 1 |
0 3 -2 0 

## The Attempt at a Solution

I have reduced the matrix into rref of
3 0 1 3
0 3-2 0
0 0 0 0
But am struggling to find the column vectors which are the solutions of that, which I believe is the kernel
And then I know that the rank and nullity are the dimensions of the image and kernel respectively

The "kernel" of matrix A is the set of all vectors v such that Av= 0. If you write vector v as (a, b, c, d) then any vector in the kernel must satisfy a- b+ c+ d= 0, a+ 2b- c+ d= 0, and 3b- 2c= 0. The "image" is the set of all vectors, v, such that Ax= v for some vector x. If we write vector x as (a, b, c, d) then v must be (x, y, z) such that a- b+ c+ d= x, a+ 2b- c+ d= y, and 3b- 2c= z for some numbers a, b ,c, and d.

## 1. What is a matrix?

A matrix is a rectangular array of numbers or symbols arranged in rows and columns. It is commonly used in mathematics, physics, computer science, and other fields to represent and manipulate data and equations.

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

The kernel of a matrix, also known as the null space, is the set of all vectors that when multiplied by the matrix result in the zero vector. In other words, it is the set of all solutions to the equation Ax = 0, where A is the given matrix. The dimension of the kernel is known as the nullity of the matrix.

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

The image of a matrix is the set of all possible outputs that can be obtained by multiplying the matrix by any valid input vector. It is also known as the column space of the matrix. To find the image, we can use row reduction techniques or the Rank-Nullity theorem, which states that the dimension of the image is equal to the rank of the matrix.

## 4. How do you determine the rank of a matrix?

The rank of a matrix is the maximum number of linearly independent columns (or rows) in the matrix. It can be determined by performing row reduction on the matrix and counting the number of non-zero rows in the reduced row-echelon form. The rank is also equal to the number of pivot columns in the matrix or the dimension of the image.

## 5. What is the relationship between the rank and nullity of a matrix?

The Rank-Nullity theorem states that the sum of the rank and nullity of a matrix is equal to the number of columns (or rows) in the matrix. In other words, the rank and nullity are complementary. This means that as the rank of a matrix increases, the nullity decreases and vice versa. The rank and nullity also provide important information about the solutions to a system of linear equations represented by the matrix.

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