# 10.3 Determine if A is in the span B

• MHB
• karush
In summary, the conversation discusses determining if a given matrix A is in the span of a set of matrices B. The process involves setting up a system of equations and solving for the coefficients. If the system has no solutions, then A is not in the span of B. If the system has a unique solution, then A is in the span of B.
karush
Gold Member
MHB
Determine if $A=\begin{bmatrix} 1\\3\\2 \end{bmatrix}$ is in the span $B=\left\{\begin{bmatrix} 2\\1\\0 \end{bmatrix} \cdot \begin{bmatrix} 1\\1\\1 \end{bmatrix}\right\}$

ok I added A and B to this for the OP
but from examples it looks like this can be answered by scalors so if

$c_1\begin{bmatrix} 2\\1\\0 \end{bmatrix} + c_2\begin{bmatrix} 1\\1\\1 \end{bmatrix}=\begin{bmatrix} 1\\3\\2 \end{bmatrix}$

Hi karush.

So you have
$$\begin{eqnarray}2c_1 &+& c_2 &=& 1 \\ c_1 &+& c_2 &=& 3 \\ {} & {} & c_2 &=& 2.\end{eqnarray}$$
If you substitute $c_2=2$ from the last equation into the first two equations, you get two different values for $c_1$. Hence the above set of equations is inconsistent (has no solutions) showing that $\mathbf A\notin\mathrm{span}B$.

Lets try this one... if $A= \begin{bmatrix} 1\\3\\2 \end{bmatrix}$ is in the span $B=\left\{\begin{bmatrix} 2\\1\\0 \end{bmatrix} \cdot \begin{bmatrix} 1\\1\\1 \end{bmatrix} \cdot \begin{bmatrix} 0\\1\\1 \end{bmatrix}\right\}$
then
$\begin{array}{rrrrr} 2c_1 &+ c_2 & & =1 \\ c_1 &+ c_2 & +c_3 & =3 \\ & c_2 & +c_3 & =2 \end{array}$
Solving $c_1=1, c_2=−1, c_3=3$
so $A\in\mathrm{span}B$

Last edited:
You are missing some "+" signs, aren't you?

Yes, the definition of "span" requires that
$2c_1+ c_2= 1$
$c_1+ c_2+ c_3= 3$ and
$c_2+ c_3= 2$

## 1. What does it mean to determine if A is in the span B?

Determining if A is in the span B means to check if vector A can be written as a linear combination of the vectors in span B. In other words, if A can be expressed as a sum of multiples of the vectors in B.

## 2. How do you determine if A is in the span B?

To determine if A is in the span B, you can use the row reduction method or the Gaussian elimination method. Both methods involve creating an augmented matrix with the vectors in B and A, and then performing elementary row operations to see if the resulting matrix has a solution. If it does, then A is in the span B.

## 3. What is the significance of determining if A is in the span B?

Determining if A is in the span B can help us understand the relationship between the vectors in B and A. It can also help us determine if a vector is a linear combination of other vectors, which is useful in many applications in physics, engineering, and other fields.

## 4. Can A be in the span B if B is an empty set?

No, A cannot be in the span B if B is an empty set. This is because the span of B is defined as the set of all linear combinations of the vectors in B. If B is empty, there are no vectors to combine and therefore A cannot be in the span B.

## 5. What are the possible outcomes of determining if A is in the span B?

There are two possible outcomes when determining if A is in the span B: either A is in the span B, meaning it can be written as a linear combination of the vectors in B, or A is not in the span B, meaning it cannot be written as a linear combination of the vectors in B. In the latter case, A may still be in the span of a different set of vectors.

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