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I'm aware that very similar questions have been asked but I could really do with someone walking me through this question.

What is the span of the set

$$\underline a = \begin{pmatrix}1 \\ -1 \\ 2 \\ 1 \end{pmatrix},\ \underline b = \begin{pmatrix}2 \\ -3 \\ 4 \\ 1 \end{pmatrix},\ \underline c = \begin{pmatrix}1 \\ 1 \\ 2 \\ 3 \end{pmatrix}$$

I know I need to find all $\underline b = \lambda_1\underline a +\lambda_2\underline b + \lambda_3\underline c$

So find the solution to the equation

$$\begin{pmatrix}1 & 2 & 1 \\ -1 & -3 & 1 \\ 2 & 4 & 2 \\ 1 & 1 & 3\end{pmatrix}\begin{pmatrix}\lambda_1 \\ \lambda_2 \\ \lambda_3 \end{pmatrix}=\begin{pmatrix}b_1 \\ b_2 \\ b_3 \\ b_4\end{pmatrix}$$

But I can't interpret the answer.

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Observe that $$ \left( \begin{array}{c} 1 \\ 1 \\ 2 \\ 3 \\ \end{array} \right) = 5\left( \begin{array}{c} 1 \\ -1 \\ 2 \\ 1 \\ \end{array} \right)-2\left( \begin{array}{c} 2 \\ -3 \\ 4 \\ 1 \\ \end{array} \right), $$ and $\underline a$ is not a scalar multiple of $\underline b$. Hence, $\operatorname{Span}(\{\underline a,\underline b,\underline c\}) = \operatorname{Span}(\{\underline a,\underline b\})$.

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The span of vectors is not really something we can find. We can graph the span of vectors or find the dimension of the span of the vectors. We can also find bases for the span of a set of vectors (in this case you already have one). However, it doesn't really make sense to "find" the span of vectors. Remember, the span of a set of vectors is the set of all possible linear combinations of the set of vectors.

Edit: Since the set of vectors is linearly dependent, you can create a basis for the span with two of the given vectors. However, you still do not solve that equation. Depending on what your professor is asking, you can provide a basis of linearly independent vectors with a span equivalent to the given set of vectors. The question not specific enough.

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  • $\begingroup$ Okay thank-you, got an exam tomorrow I think I'm just getting myself confused :C $\endgroup$ Commented Jan 24, 2020 at 1:29
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    $\begingroup$ Good luck! If you're asked to sketch the span of a set of vectors, just add up the vectors with different weights in any way possible and sketch all possible results. $\endgroup$ Commented Jan 24, 2020 at 1:32
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All that you should be looking for is the reduced echelon form (REF) of the matrix: $$\begin{pmatrix}1&2&1\\ \:\:-1&-3&1\\ \:\:2&4&2\\ \:\:1&1&3\end{pmatrix}$$ Which becomes by applying Gaussian elimination: $$\begin{pmatrix}1&2&1\\ 0&1&-2\\ 0&0&0\\ 0&0&0\end{pmatrix}$$ Now the basis of the column space of this matrix is also the basis of the span of the three vectors, and to find it we simply take the vectors in the original matrix which correspond to the columns in the REF that have a leading one: $$\begin{pmatrix}1\\ -1\\ 2\\ 1\end{pmatrix}\:,\:\begin{pmatrix}2\\ \:-3\\ \:4\\ \:1\end{pmatrix}$$

These are the two vectors that you should be looking for.

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