A4 - Surjective and Injective Transformations


Example 1

A4 - Surjective and Injective Transformations (ver. 1)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 1 & -6 & 8 \\ 0 & 1 & -5 & 6 \\ -2 & 1 & 4 & -5 \\ 0 & -1 & 2 & -3 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 2

A4 - Surjective and Injective Transformations (ver. 2)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} -2 & -6 & -5 & 1 \\ 2 & 1 & 6 & 8 \\ 0 & 0 & 1 & 2 \\ 1 & 2 & 3 & 2 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

The transformation is surjective and injective.


Example 3

A4 - Surjective and Injective Transformations (ver. 3)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} -2 & 1 & -3 \\ 3 & -8 & -2 \\ 1 & 2 & 4 \\ 3 & -4 & 2 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 2 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 4

A4 - Surjective and Injective Transformations (ver. 4)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & -1 & -1 & 2 \\ 4 & -3 & -2 & 2 \\ 3 & -3 & -2 & 1 \\ -2 & 1 & 1 & -2 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

The transformation is surjective and injective.


Example 5

A4 - Surjective and Injective Transformations (ver. 5)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & -2 \\ -3 & 4 & -7 \\ 3 & -4 & 7 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and injective.


Example 6

A4 - Surjective and Injective Transformations (ver. 6)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 1 & -3 & -2 \\ -5 & -4 & 0 & -5 \\ 0 & 0 & 1 & 1 \\ 3 & 4 & -6 & -3 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 7

A4 - Surjective and Injective Transformations (ver. 7)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & -1 & -3 & -5 \\ 2 & -1 & -2 & -3 \\ -2 & 1 & 3 & 4 \\ 1 & 0 & -2 & -1 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 8

A4 - Surjective and Injective Transformations (ver. 8)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 1 & -3 & -2 \\ -3 & -2 & 5 & -1 \\ -2 & -2 & 7 & 6 \\ -3 & -1 & 4 & -2 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 9

A4 - Surjective and Injective Transformations (ver. 9)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} -2 & 0 & -3 & 4 \\ -2 & 1 & 5 & 1 \\ 1 & 0 & 1 & -2 \\ 2 & 1 & 8 & -6 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

The transformation is surjective and injective.


Example 10

A4 - Surjective and Injective Transformations (ver. 10)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 0 & 1 & 0 & 5 \\ -1 & 1 & 3 & -2 \\ 0 & 2 & 1 & 6 \\ 0 & 3 & 2 & 8 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

The transformation is surjective and injective.


Example 11

A4 - Surjective and Injective Transformations (ver. 11)

Let \(T:\mathbb R^{4}\to\mathbb R^{3}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 4 & 3 & -5 & 1 \\ 2 & 1 & -2 & 0 \\ -1 & -1 & 1 & 1 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -3 \end{array}\right]\]

The transformation is surjective and not injective.


Example 12

A4 - Surjective and Injective Transformations (ver. 12)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & 2 & 7 \\ 0 & 1 & 4 \\ 0 & -1 & -3 \\ 0 & 0 & -2 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and injective.


Example 13

A4 - Surjective and Injective Transformations (ver. 13)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & 0 & -4 \\ 0 & 1 & -2 \\ 0 & 1 & -1 \\ -3 & 0 & 7 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and injective.


Example 14

A4 - Surjective and Injective Transformations (ver. 14)

Let \(T:\mathbb R^{4}\to\mathbb R^{3}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 4 & -3 & -5 \\ -1 & -4 & 4 & 7 \\ -1 & -4 & 4 & 7 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 4 & 0 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 15

A4 - Surjective and Injective Transformations (ver. 15)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 1 & -3 & -3 \\ 2 & -1 & -8 & 6 \\ 0 & -1 & -1 & 4 \\ 1 & 0 & -5 & 1 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 16

A4 - Surjective and Injective Transformations (ver. 16)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} -2 & -1 & 6 \\ -3 & -2 & 7 \\ 1 & 1 & 0 \\ -1 & 2 & 8 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and injective.


Example 17

A4 - Surjective and Injective Transformations (ver. 17)

Let \(T:\mathbb R^{4}\to\mathbb R^{3}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 2 & 5 & 7 \\ -2 & -3 & -6 & -8 \\ 0 & 1 & 5 & 8 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 2 \end{array}\right]\]

The transformation is surjective and not injective.


Example 18

A4 - Surjective and Injective Transformations (ver. 18)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 0 & 2 & -2 \\ 0 & 1 & -4 & 5 \\ 1 & 0 & 3 & -2 \\ -1 & 0 & -6 & 3 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

The transformation is surjective and injective.


Example 19

A4 - Surjective and Injective Transformations (ver. 19)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} -1 & 1 & 5 \\ 2 & -1 & -8 \\ 0 & 0 & 0 \\ 1 & -1 & -5 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 20

A4 - Surjective and Injective Transformations (ver. 20)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & 1 & -5 \\ 1 & 2 & -8 \\ -2 & 0 & 4 \\ 0 & -2 & 6 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & -2 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 21

A4 - Surjective and Injective Transformations (ver. 21)

Let \(T:\mathbb R^{4}\to\mathbb R^{3}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 3 & -4 & -6 & 0 \\ -2 & 1 & 5 & -1 \\ -2 & 4 & 3 & 1 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \end{array}\right]\]

The transformation is surjective and not injective.


Example 22

A4 - Surjective and Injective Transformations (ver. 22)

Let \(T:\mathbb R^{4}\to\mathbb R^{3}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & -2 & -7 & -4 \\ 1 & -1 & -3 & -1 \\ -1 & 1 & 4 & 1 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 0 \end{array}\right]\]

The transformation is surjective and not injective.


Example 23

A4 - Surjective and Injective Transformations (ver. 23)

Let \(T:\mathbb R^{4}\to\mathbb R^{3}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 0 & 1 & -4 \\ 3 & 1 & 1 & -5 \\ 2 & 1 & 1 & -4 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -3 \end{array}\right]\]

The transformation is surjective and not injective.


Example 24

A4 - Surjective and Injective Transformations (ver. 24)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & -1 & 5 \\ 0 & 1 & -1 \\ -2 & -5 & -2 \\ 0 & 5 & -8 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and injective.


Example 25

A4 - Surjective and Injective Transformations (ver. 25)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} -1 & -5 & -5 \\ 0 & 0 & 1 \\ -1 & -5 & 0 \\ -1 & -5 & -1 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 5 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 26

A4 - Surjective and Injective Transformations (ver. 26)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 0 & 1 & -3 & -5 \\ 0 & 1 & 0 & 1 \\ 1 & -1 & 0 & 0 \\ -1 & 6 & -2 & 1 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 27

A4 - Surjective and Injective Transformations (ver. 27)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & 1 & 0 \\ 1 & -2 & 6 \\ 0 & 2 & -4 \\ 0 & -1 & 2 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 2 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 28

A4 - Surjective and Injective Transformations (ver. 28)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} -1 & 1 & -2 \\ -1 & 0 & -4 \\ 1 & 0 & 4 \\ 2 & -1 & 6 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 4 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 29

A4 - Surjective and Injective Transformations (ver. 29)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & -5 & -2 & 7 \\ 0 & 1 & 1 & -3 \\ 0 & 1 & 2 & -6 \\ 0 & -2 & 0 & 0 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 30

A4 - Surjective and Injective Transformations (ver. 30)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & 0 & 2 \\ 2 & 1 & -1 \\ -4 & 0 & -7 \\ -2 & -1 & 4 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and injective.


Example 31

A4 - Surjective and Injective Transformations (ver. 31)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & -1 & -1 \\ 5 & -4 & -6 \\ 5 & -5 & -5 \\ 1 & -6 & 4 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & -2 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 32

A4 - Surjective and Injective Transformations (ver. 32)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 4 & 1 & 7 & -5 \\ -1 & 0 & -3 & 0 \\ 1 & 0 & 4 & 1 \\ 1 & 0 & 3 & 0 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 33

A4 - Surjective and Injective Transformations (ver. 33)

Let \(T:\mathbb R^{4}\to\mathbb R^{3}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & -2 & -1 & 5 \\ 1 & -1 & 0 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 34

A4 - Surjective and Injective Transformations (ver. 34)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & -5 & -2 & -8 \\ -1 & 6 & 3 & 8 \\ -1 & 5 & 3 & 5 \\ 0 & -5 & -6 & 3 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 35

A4 - Surjective and Injective Transformations (ver. 35)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & 1 & 6 \\ 0 & 1 & 3 \\ 0 & 1 & 3 \\ 1 & -1 & 0 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 3 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 36

A4 - Surjective and Injective Transformations (ver. 36)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 1 & 0 & -5 \\ -3 & -2 & 1 & 8 \\ -1 & -2 & 0 & 7 \\ -2 & -2 & 1 & 6 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

The transformation is surjective and injective.


Example 37

A4 - Surjective and Injective Transformations (ver. 37)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 5 & 3 & -4 & 1 \\ -2 & -1 & 1 & 1 \\ -4 & -3 & 6 & -7 \\ 1 & 1 & -2 & 3 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 38

A4 - Surjective and Injective Transformations (ver. 38)

Let \(T:\mathbb R^{4}\to\mathbb R^{3}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 0 & 2 & -5 \\ 4 & 1 & 1 & 6 \\ 0 & 0 & 1 & -4 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -4 \end{array}\right]\]

The transformation is surjective and not injective.


Example 39

A4 - Surjective and Injective Transformations (ver. 39)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 4 & 3 & -8 & 5 \\ -3 & -2 & 6 & -5 \\ 4 & 3 & -7 & 2 \\ 5 & 5 & -7 & -8 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

The transformation is surjective and injective.


Example 40

A4 - Surjective and Injective Transformations (ver. 40)

Let \(T:\mathbb R^{4}\to\mathbb R^{3}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 0 & 3 & -5 \\ 0 & 1 & 2 & -6 \\ -1 & 1 & 0 & -3 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -2 \end{array}\right]\]

The transformation is surjective and not injective.


Example 41

A4 - Surjective and Injective Transformations (ver. 41)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & 6 & 7 \\ 0 & 1 & 1 \\ 1 & 5 & 6 \\ -1 & -5 & -6 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 42

A4 - Surjective and Injective Transformations (ver. 42)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & -4 & 8 \\ 1 & -3 & 6 \\ -1 & 3 & -6 \\ 0 & 0 & 0 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 43

A4 - Surjective and Injective Transformations (ver. 43)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & -1 & -3 \\ 2 & -1 & -2 \\ -4 & 2 & 5 \\ -1 & 1 & 8 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and injective.


Example 44

A4 - Surjective and Injective Transformations (ver. 44)

Let \(T:\mathbb R^{4}\to\mathbb R^{3}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & -1 \\ 1 & 1 & -1 & 1 \\ 0 & 0 & 1 & -2 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -2 \end{array}\right]\]

The transformation is surjective and not injective.


Example 45

A4 - Surjective and Injective Transformations (ver. 45)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 2 & -4 & 8 \\ -1 & -1 & 2 & -3 \\ 0 & 2 & -3 & 7 \\ -1 & 0 & 2 & -4 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 46

A4 - Surjective and Injective Transformations (ver. 46)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & 1 & 5 \\ 2 & 1 & 6 \\ -1 & 1 & -8 \\ 0 & -1 & 0 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and injective.


Example 47

A4 - Surjective and Injective Transformations (ver. 47)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 1 & -3 & 2 \\ 1 & 2 & -5 & 3 \\ 5 & 2 & -8 & 4 \\ 2 & 2 & -4 & -1 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

The transformation is surjective and injective.


Example 48

A4 - Surjective and Injective Transformations (ver. 48)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrr} 1 & -2 & 6 \\ 0 & 1 & -2 \\ 0 & -3 & 6 \\ -1 & 2 & -6 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrr} 1 & 0 & 2 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The transformation is not surjective and not injective.


Example 49

A4 - Surjective and Injective Transformations (ver. 49)

Let \(T:\mathbb R^{4}\to\mathbb R^{3}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 0 & 1 & 0 & 1 \\ -1 & 4 & -4 & 7 \\ -1 & -2 & -3 & 0 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 \end{array}\right]\]

The transformation is surjective and not injective.


Example 50

A4 - Surjective and Injective Transformations (ver. 50)

Let \(T:\mathbb R^{4}\to\mathbb R^{4}\) be the transformation given by the standard matrix

\[\left[\begin{array}{rrrr} 1 & 0 & -2 & 5 \\ -1 & 1 & 3 & -8 \\ 3 & 2 & -3 & 7 \\ -1 & -1 & 4 & -7 \end{array}\right]\]

(a) Explain why \(T\) is or is not surjective.

(b) Explain why \(T\) is or is not injective.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

The transformation is surjective and injective.