For the matrices (a) find k such that Nul A

arneguet9k 2021-11-17 Answered
For the matrices (a) find k such that Nul A is a subspace of \(\displaystyle{\mathbb{{{R}}}}^{{k}}\), and (b) find k such that Col A is a subspace of \(\displaystyle{\mathbb{{{R}}}}^{{k}}\)
\[A=\begin{bmatrix}4&5&-2&6&0\\1&1&0&1&0\end{bmatrix}\]

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Expert Answer

Feas1981
Answered 2021-11-18 Author has 8615 answers
a) In order to find Nul A, we must solve
\(\displaystyle{A}{x}={0}\)
Since A is \(\displaystyle{2}\times{5}\) matrix, it can multyplied from left only by vector with 5 rows. That means x is in \(\displaystyle{\mathbb{{{R}}}}^{{5}}\), i.e. k=5
b) As we saw in part a) \(\displaystyle{x}\in{\mathbb{{{R}}}}^{{5}}\), so we know
Col \(\displaystyle{A}={\left\lbrace{y}:{y}={A}{x}\ \text{ for some }\ {x}\in{\mathbb{{{R}}}}^{{5}}\right\rbrace}\)
So we have a multiplication of \(\displaystyle{2}\times{5}\) matrix A and \(\displaystyle{5}\times{1}\) vector x. From definition of matrix muplication, y is \(\displaystyle{2}\times{1}\) vector, i.e k=2
Result: a) 5, b) 2
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