babeeb0oL

2020-12-15

Find and describe an example of a matrix with each of the following properties.

Briefly describe why the desired property is in the matrix you picked. If no such matrix exists, explain using a theorem studied.

(a) A square matrix representing an injective but not a surjective transformation.

Briefly describe why the desired property is in the matrix you picked. If no such matrix exists, explain using a theorem studied.

(a) A square matrix representing an injective but not a surjective transformation.

smallq9

Skilled2020-12-16Added 106 answers

Find and describe an example of a matrix with each of the following properties.

(a) A square matrix representing an injective but not a surjective transformation.

Given: we have a square matrix representing an injective transformation.

Explanation: in general for an$m\times n$ matrix A the rank

rank$A=minm,n$

matrix A will be

1) injective if$m\ge n=\text{}rank\text{}A\Rightarrow Dim(kerA)=0$

2) surjective if$n\ge m=$ rank A

3) bijective if$m=n=$ rank A

Now injective but not surjective let us consider an example .

${A}_{3\times 2}=\left[\begin{array}{c}20\\ 03\\ 00\end{array}\right]$

here in the above matrix the number of rows are more than the number of column that is

$3>2=$ rank A

since A is injective

but rank$A\ne 3=$ dimension of the codomain

hence A is not surjective

(a) A square matrix representing an injective but not a surjective transformation.

Given: we have a square matrix representing an injective transformation.

Explanation: in general for an

rank

matrix A will be

1) injective if

2) surjective if

3) bijective if

Now injective but not surjective let us consider an example .

here in the above matrix the number of rows are more than the number of column that is

since A is injective

but rank

hence A is not surjective

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