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ksaimouli
 one year ago
Linear algebra question
ksaimouli
 one year ago
Linear algebra question

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ksaimouli
 one year ago
Best ResponseYou've already chosen the best response.0\[\left[\begin{matrix}1 & 0&4 \\0 & 3&2 \\ 2 & 6& 3\end{matrix}\right]\]

ksaimouli
 one year ago
Best ResponseYou've already chosen the best response.0Let ^ be A, b= \[\left(\begin{matrix}4 \\ 1\\ 4\end{matrix}\right)\]

ksaimouli
 one year ago
Best ResponseYou've already chosen the best response.0columns of A is a1,a2,a3 and W= span{a1,a2,a3}

ksaimouli
 one year ago
Best ResponseYou've already chosen the best response.0Is b in W? how many vectors are in {a1,a2,a3}

ksaimouli
 one year ago
Best ResponseYou've already chosen the best response.0If W is in span of A, means C1a1+C2a2+C3a3=W?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Looks like you're just checking to see if \(b\) can be written as a linear combination of \(a_1,a_2,a_3\), which amounts to solving for \(c_1,c_2,c_3\) such that \[c_1a_1+c_2a_2+c_3a_3=\begin{pmatrix}b_1\\b_2\\b_3\end{pmatrix}\] which means you'll be solving a system of 3 equations with 3 unknowns.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0In other words, \[\begin{cases} c_12c_3=4\\ 3c_2+6c_3=1\\ 4c_12c_2+3c_3=4 \end{cases}\]

ksaimouli
 one year ago
Best ResponseYou've already chosen the best response.0True, I found it to be unique. Which means b is in the span of W. How many vectors are in W?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Well, I'd say there's an infinite number of vectors. \(a_1,a_2,a_3\) are linearly independent, so \(W\) forms a basis of \(\mathbb{R}^3\), which contains an infinite number of vectors.

ksaimouli
 one year ago
Best ResponseYou've already chosen the best response.0I see, how would be interpret if the question was Is W in b?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I don't think that question would make sense. \(W\) is a set/space of vectors, while \(b\) is just a vector. A set can't belong to an element.

ksaimouli
 one year ago
Best ResponseYou've already chosen the best response.0Okay, you said "number of vectors. a1,a2,a3 are linearly independent" is it because they are not scalar multiplies of each other?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Yes, the only way to get \(k_1a_1+k_2a_2+k_3a_3=\begin{pmatrix}0\\0\\0\end{pmatrix}\) is if all the \(k=0\). It's more accurate to say "linear combination" in place of "scalar multiples" here. (The second term is a special case of the first term.)

ksaimouli
 one year ago
Best ResponseYou've already chosen the best response.0thank you so much :)
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