# The span of Vectors Calculator + Online Solver With Free Steps

A Span of Vectors Calculator is a simple online tool that computes the set of all linear combinations of two vectors or more.

By employing this calculator, you can consistently show the distribution of a vector function.

## What Is a Span of Vectors Calculator?

The Span of Vectors Calculator is a calculator that returns a list of all linear vector combinations. For instance, if $v_1 = [11,5,-7,0]^T$  and $v_1 = [2,13,0,-7]^T$, the set of all vectors of the form $s \cdot v^1+t \cdot v^2$ for certain scalars ‘s’ and ‘t’ is the span of v1 and v2.

A subspace of $\mathbb{R}^n$ is given by the span of a set of vectors in that space. Any non-trivial subdomain can be expressed as the span of any one of an infinite number of vector set combinations.

If λi = 0 exists as the only solution to the vector expression {λ1.V1 +…..+ λm.Vm}, a collection of vectors {V1, . . . , Vm} are linearly independent. It is only linearly dependent when a series of vectors are not linearly independent.

The rows of A, for instance, are not linearly independent because

$-\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix} + \begin{bmatrix} -2 \\ 3 \\ -1 \\ 5 \end{bmatrix} + \begin{bmatrix} 3 \\ -1 \\ 4 \\ -1 \end{bmatrix} = 0$

To check if a group of vectors is linearly independent or not, represent them as columns of a matrix C and compute Cx=0.
The vectors are linearly dependent if there are any nontrivial solutions; else, these vectors are linearly independent.

## How To Use a Span of Vectors Calculator

You can use the calculator by carefully following the step-by-step instructions below; you can use the Span of Vectors Calculator to get the desired results. Therefore, you can adhere to the guidelines to obtain the desired result.

### Step 1

Enter the values of Vector 1 and Vector 2 in the provided entry boxes.

### Step 2

Press the “Span Me” button to calculate the Span of Vectors for the given vectors and to view the detailed, step-by-step solution for the Span of Vector Calculation.

## How Does a Span of Vectors Calculator Work?

The Span of Vectors Calculator works by determining all the possible linear combinations of multiple vectors.

If S is linearly dependent, given a group of vectors S ={v1, v2, … , vn}, then zero is a significant linear combination of vectors in S. That is, if  and only if $c_1 \cdot v_1 + c_2 \cdot v_2 +… + c_n \cdot v_n = 0$, then there are constants { c1,…, cn } with at least one of the constants nonzero.

If S is assumed to be linearly dependent, then:

$v_i = c_1 \cdot v_1 + c_2 \cdot v_2 + … + c_{i-1} \cdot v_{i-1} + c_{i+1} \cdot v_{i+1} + … + c_n \cdot v_n$

$V_i$ is subtracted from both sides to give us:

$c_1 \cdot v_1 + c_2 \cdot v_2 + … + c_{i-1} \cdot v_{i-1} + c_{i+1} \cdot v_{i+1} + … + c_n \cdot v_n = 0$

The nonzero value of ci in the equation above causes 0 to be a nontrivial linear combination of vectors in S.

Let’s now consider:

$c_1 \cdot v_1 + c_2 \cdot v_2 + … + c_{i-1} \cdot v_{i-1} + c_i \cdot v_i + c_{i+1} \cdot v_{i+1} + … + c_n \cdot v_n = 0$

With nonzero ci. Let $a_j = – \frac{c_j}{c_i}$ be the result from multiplying both sides of the equation by ci:

$-a_1 v_1 – a_2 v_2 – … – a_{i-1} v_{i-1} + v_i- a_{i+1} v_{i+1} – … – a_n v_n = 0$

Lastly, reposition each term to the right side of the equation:

$vi = a_1 v_1 + a_2 v_2 + … + a_{i-1} v_{i-1} + v_i + a_{i+1} v_{i+1} + … + a_n v_n$

The line across the origin determined by x1 is the span of a single nonzero vector x1 in R3 (or R2).

The collection of all x1’s potential linear combinations, or all x1’s of the type 1×1, where $\alpha \cdot 1 \in \mathbb{R}$, is known as spam. The line across the origin given by x1 is called the span of x1, which is the set of all multiples of x1.

### List of Some Linear Combinations

Here are some examples of vector combinations:

Vector 0: span(0) equals 0.

span(v) = 1 vector, which is a line.

If two vectors v1 and v2 are not collinear, then span(v1, v2) = $\mathbb{R}^2$.

span(v1, v2, v3…) = $\mathbb{R}^2$ for three or more vectors. All vectors, excluding two, are redundant.

## Solved Examples

Let’s explore some examples better to understand the working of the Vector Function Grapher Calculator.

### Example 1

Demonstrate that the set

S =  {(1, 1, 0), (0, 1, 1), (1, 1, 2)}

spans $\mathbb{R}^3$ and represents the vector (2,4,8) as a linear combination of vectors in S.

### Solution

A vector in $\mathbb{R}^3$ has the following form:

v  =  (x, y, z)

Therefore, we must demonstrate that every such v may be expressed as:

$(x,y,z) = c_1(1, 1, 0) + c_2(0, 1, 1) + c_3(1, 1, 2)$

$(x,y,z) = (c_2 + c_3, c_1 + c_3, c_1 + c_2)$

This is compatible with the set of equations:

$c_2 + c_3 = x$

$c_1 + c_3 = y$

$c_1 + c_2 = z$

It can be expressed as a matrix:

$\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 2 \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$

We can write this as:

A . c  =  b

Notice that:

det(A)  =  2

Hence A is nonsingular and:

c  =  A$^{-1}$ . b

So there is a nontrivial solution. We discover that (2,4,8) can be written as a linear combination of vectors in S.

$A^{-1} = \begin{bmatrix} .5 & -1 & .5 \\ .5 & 1 & -.5 \\ -.5 & 0 & -.5 \end{bmatrix}$

Then:

$c = \begin{bmatrix} .5 & -1 & .5 \\ .5 & 1 & -.5 \\ -.5 & 0 & -.5 \end{bmatrix} \begin{bmatrix} 2 \\ 4 \\ 8 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix}$

We have:

(2,4,8)  =  1(0,1,1) + 1(1,0,1) + 3(1,1,0)

### Example 2

Show that S does not span $P_2$ if $v_1 = t + 2$ and $v_2 = t_2 + 1$ and $S = {v_1, v_2}$.

### Solution

A general element of $P_2$ is of the form

$v = at^2 + bt + c$

We set

$v = c_1 v_11 + c_2 v_2$

or

$at^2 + bt + c = c_2 (t + 2) + c_2 (t^2 + 1) = c_2 \cdot t^2 + c_1 \cdot t + c_1 + c_2$

Equating coefficients gives

a  =  c1

b  =  c1

c  =  c1 + c2

Notice that if:

a  =  1    b  =  1    c  =  1

There is no solution to this. Hence, ‘S’ does not span ‘V.’