How can we find the span of the two vectors $v_1=\left(\begin{smallmatrix}3\\-4\\5\\7\end{smallmatrix}\right)$ and $v_2=\left(\begin{smallmatrix}2\\1\\-5\\-5\end{smallmatrix}\right)$ in $\mathbb{R}^4$ ?
How can we find the span of two vectors in $\mathbb{R}^4$?
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linear-algebra
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1It's the set of all possible linear combinations of $v_1$ and $v_2$ by definition. – 2017-02-26
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0Do you mean by c1v1+c2v2 =0? – 2017-02-26
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0I mean it is the set $S=\{c_1v_1+c_2v_2 : c_1,c_2 \in \mathbb{R} \}$ – 2017-02-26
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3 Answers
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The span of two vectors $v_1$ and $v_2$ is given by the set,
$\{\alpha v_1 + \beta v_2: \alpha, \beta \in \mathbb{R}\}$ which could be simplified as
$(3 \alpha+2 \beta,−4 \alpha +1 \beta ,5 \alpha −5 \beta ,7 \alpha −5 \beta)$
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I'm not quite sure what you mean by finding the span of the two vectors in $\mathbb{R}^4$, but if you're asking what the two vectors span in $\mathbb{R}^4$, it would only be a two-dimensional plane.
It's only a two-dimensional plane because you have two vectors which are linearly independent. We know that they're linearly independent because you cannot write one as a scalar multiple of the other, even if said scalar is a negative value.
Therefore, we can form any vector that lies in a two-dimensional plane in $\mathbb{R}^4$ with the equation:
$\underline{v}_x=c_1\underline{v}_1+c_2\underline{v}_2 \qquad \qquad c_1,c_2 \in \mathbb{R}$
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The span of the two vectors $v_1=\left(\begin{smallmatrix}3\\-4\\5\\7\end{smallmatrix}\right)$ and $v_2=\left(\begin{smallmatrix}2\\1\\-5\\-5\end{smallmatrix}\right)$ would be the set of vectors $\left(\begin{smallmatrix}x\\y\\z\\w\end{smallmatrix}\right)$ such that $$\pmatrix{x\\y\\z\\w}=av_1 + bv_2=a\pmatrix{3\\-4\\5\\7}\:+\:b\pmatrix{2\\1\\-5\\-5},$$ that is the set of all vectors of the form $\left(\begin{smallmatrix}3a+2b\\-4a+b\\5a-5b\\7a-5b\end{smallmatrix}\right)$, which as @Ahmed said represents the set of all linear combinations of $v_1$ and $v_2$.