# Linear Algebra - Kernel and range of T

• SetepenSeth
In summary, the linear transformation T is defined on the space of 2x2 matrices and maps the matrix to a diagonal matrix with a and d as the diagonal entries. The range of T is spanned by the matrices [1 0; 0 0] and [0 0; 0 1], which also form a basis for the range. The kernel of T is the set of all matrices where a and d are equal to 0, and the dimension of the kernel is 2. The dimension of the space of 2x2 matrices is 4.
SetepenSeth

## Homework Statement

Let ##T:M_2 \to M_2## a linear transformation defined by

##T \begin{bmatrix}
a&b\\
c&d
\end{bmatrix} =
\begin{bmatrix}
a&0\\
0&d
\end{bmatrix}##

Describe ##ker(T)## and ##range(T)##, and find their basis.

## Homework Equations

For a linear transformation ##T:V\to W##

##range(T)={T(x) \epsilon W : x \epsilon V}##

##ker(T)= {x \epsilon V : T(x)= 0 \epsilon W}##

## The Attempt at a Solution

Skipping the first part of the proof, I get to the part where I describe the range of the transformation and express it as a linear combination of two ##M_2## matrix

##T \begin{bmatrix}
a&b\\
c&d
\end{bmatrix} =
\begin{bmatrix}
a&0\\
0&d
\end{bmatrix}##

##T \begin{bmatrix}
a&b\\
c&d
\end{bmatrix} =
a\begin{bmatrix}
1&0\\
0&0
\end{bmatrix}+
d\begin{bmatrix}
0&0\\
0&1
\end{bmatrix}
##

So $$\begin{bmatrix} 1&0\\ 0&0 \end{bmatrix} and \begin{bmatrix} 0&0\\ 0&1 \end{bmatrix}$$ span the range for ##T##, also they are linearly independent, thus forming a basis for the range.

The kernel can be expressed as

##ker(T)={A \epsilon M_2 : Ax=0 \epsilon M_2}##

##ker(T)=A \epsilon M_2 :## ##A =
\begin{bmatrix}
a&b\\
c&d
\end{bmatrix} \forall a,b,c,d \epsilon ℝ, a=d=0## If I got it right up to this point then

##dim[range(T)]=2##

But then according to the theorem that says

##dim[range(T)]+dim[ker(T)]=dim(V)##

Being ##V## the space of ##2_x####2## square matrix, then ##dim(V)=2## but that would make ##dim[ker(T)]=0## which doesn't make sense to me, so I believe I got a concept wrong somewhere on my analysis.

Why do you think the dimension of V is 2?

LCKurtz said:
Why do you think the dimension of V is 2?

I see, I just noticed I followed a wrong assumption

##V=
a\begin{bmatrix}
1&0\\
0&0
\end{bmatrix}+
b\begin{bmatrix}
0&1\\
0&0
\end{bmatrix}+
c\begin{bmatrix}
0&0\\
1&0
\end{bmatrix}+
d\begin{bmatrix}
0&0\\
0&1
\end{bmatrix}##

Thus ##dim(V)=4##, right?

SetepenSeth said:
I see, I just noticed I followed a wrong assumption

##V=
a\begin{bmatrix}
1&0\\
0&0
\end{bmatrix}+
b\begin{bmatrix}
0&1\\
0&0
\end{bmatrix}+
c\begin{bmatrix}
0&0\\
1&0
\end{bmatrix}+
d\begin{bmatrix}
0&0\\
0&1
\end{bmatrix}##

Thus ##dim(V)=4##, right?

Yes, in general, if you consider the vector space ##M_{m,n}(\mathbb{K})## (the ##m \times n## matrices), it has dimension ##mn##. This can easily be proven by showing that the set ##\{E_{ij}|1 \leq i \leq m, 1 \leq j \leq n\}## defines a basis for this space. Here, ##E_{ij}## is the matrix that is everywhere zero, except on place ##(i,j)## where it is ##1## (or another non zero number).

Yes.

Thank you both, now it all makes sense.

## 1. What is the kernel of a linear transformation?

The kernel of a linear transformation, also known as the null space, is the set of all input vectors that are mapped to the zero vector by the transformation. In other words, it is the set of vectors that produce a zero output when multiplied by the transformation matrix.

## 2. How is the kernel related to the range of a linear transformation?

The kernel and the range are complementary subspaces that together span the entire vector space. The range of a linear transformation is the set of all output vectors that can be obtained by multiplying the transformation matrix with any input vector from the vector space. In other words, it is the span of the columns of the matrix. The dimension of the kernel and the dimension of the range add up to the dimension of the vector space.

## 3. Can the kernel and range of a linear transformation be empty?

Yes, the kernel and range of a linear transformation can be empty. This means that the transformation matrix maps all input vectors to the zero vector, and the range is limited to just the zero vector. In this case, the transformation is called a zero transformation.

## 4. How can we determine the kernel and range of a linear transformation?

To determine the kernel of a linear transformation, we can set up a system of linear equations using the transformation matrix and solve for the variables that produce a zero output. The solutions to this system of equations will give us the basis for the kernel. To determine the range, we can use the columns of the transformation matrix and perform row reduction to find the linearly independent columns. The span of these columns will give us the basis for the range.

## 5. What is the significance of the kernel and range in linear algebra?

The kernel and range of a linear transformation play a crucial role in understanding the properties and behavior of the transformation. They provide insight into the linearity, invertibility, and dimensionality of the transformation. Additionally, they are essential in applications such as image processing, data compression, and solving systems of linear equations.

• Calculus and Beyond Homework Help
Replies
1
Views
161
• Calculus and Beyond Homework Help
Replies
19
Views
469
• Calculus and Beyond Homework Help
Replies
3
Views
216
• Calculus and Beyond Homework Help
Replies
10
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
648
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
28
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
2K
• Calculus and Beyond Homework Help
Replies
9
Views
1K