Virtual Labs

1. Add the two

5

-length binary tuples

\left(1,0,1,0,1,1\right)

and

\left(1,0,0,1,1,0\right)

over

\mathbb{F}_2

\left(2,0,1,1,2,1\right)

Explanation

\left(0,0,1,1,0,1\right)

Explanation

\left(1,0,0,0,1,0\right)

Explanation

\left(0,0,0,1,0,0\right)

Explanation

2. Consider two 3x3 matrices

A = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix}

and

B = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 1 \end{bmatrix}

. The result of matrix multiplication

AB

over

\mathbb{F}_2

\begin{bmatrix} 2 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}

Explanation

\begin{bmatrix} 0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 0 \end{bmatrix}

Explanation

\begin{bmatrix} 0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}

Explanation

\begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \end{bmatrix}

Explanation

3. Add the four

3

-length binary tuples

\left(1,0,1\right)

\left(0,0,1\right)

\left(1,0,0\right)

and

\left(0,1,1\right)

over

\mathbb{F}_2

\left(0,0,0\right)

Explanation

\left(0,0,1\right)

Explanation

\left(0,1,1\right)

Explanation

\left(0,1,0\right)

Explanation

4. The list of vectors in

\mathbb{F}_{2}^{5}

given in the set

S=\{\left(1,0,1,0,1\right), \left(0,1,0,1,1\right), \left(1,1,0,1,1\right)\}

are linearly independent. Consider the statements below, and select the one which is true.

a: If we add the vector

\left(1,0,0,0,0\right)

S

and get a new set with

4

vectors named

S'

, then the set

S'

is a linear independent set of vectors. Explanation

Explanation

b: If we add the vector

\left(0,1,1,1,0\right)

S

and get a new set with

4

vectors named

S'

, then the set

S'

is a linear independent set of vectors. Explanation

Explanation

c: If we add the vector

\left(0,0,1,0,0\right)

S

and get a new set with

4

vectors named

S'

, then the set

S'

is a linear dependent set of vectors. Explanation

Explanation

d: If we add the vector

\left(1,0,1,0,1\right)

S

and get a new set with

4

vectors named

S'

, then the set

S'

is a linear dependent set of vectors. Explanation

Explanation

5. Find the rank of matrix

R = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 \end{bmatrix}

a: 4 Explanation

Explanation

b: 3 Explanation

Explanation

c: 2 Explanation

Explanation

d: 5 Explanation

Explanation

Review of Block Codes