#3 vectors

1 messages · Page 1 of 1 (latest)

frigid oyster
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u = (2, 0, 1), w = (1, -1, 0), v_t = (t, 2, 3) and they belong to V = R^3

A is a matrix that has those 3 vectors as columns

How do I find out whether (v, w, v_t) is a base of V upon t varying in value or not. If I can't find that out, how do I find a linear dependency among those vectors

fringe heraldBOT
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frigid oyster
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t varies in R

mighty tangle
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Considering the linear (in)dependence of {u, w, v_t}
is just considering whether the linear system au + bw + cv_t = 0 has nontrivial solutions

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Hence whether A is nonsingular

frigid oyster
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but with the t as a variable can I consider that?

pliant field
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one of the approach would be to calculate determinant since it is 3x3 its not that difficult and solve det(A) ≠ 0

frigid oyster
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that t is what's throwing me off

pliant field
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since det(A) ≠0 <=> matrix has linearly independent columns

frigid oyster
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how do I also find whether a linear application f: V -> V exists so f(u) = w, f(w) = u and f(v_0) = f(v_1)

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and say that it's unique

pliant field
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if we have linear transformation f(v) = Av is where A is invertible matrix then we have bijective function and we can define f^-1

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since f is bijective function we have 1-1 correspondence which implies that f(v) is unique

frigid oyster
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oo

frigid oyster
pliant field
frigid oyster
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so if t is not 4, then those vectors are a base of V as they also generate it

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while if it's 4, what would be the linear dependency

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a would be 2 times c and b would be -3 times c

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I guess

pliant field
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to show that vectors are dependent you can show
$$ c_1 * v_1 + c_2 * v_2 + ... c_n * v_n= 0 $$ where there exists one c_i is not 0

frigid oyster
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mhm

mellow notchBOT
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L
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pliant field
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lol im still learning some latex

frigid oyster
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you're good

pliant field
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if there is some constant c_i ≠ 0 then vectors are dependent

frigid oyster
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I put them in a system and basically if t = 4, it comes out as

(t - 4)c = 0
b = 2c
a = -3c

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so then c can have any value and a,b would be dependent on it

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if you sum u and w then they are linearly dependent on v_4

pliant field
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one sec brb

pliant field
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we get linear dependence

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-3•u + 2•w + 1•v_4 = 0

frigid oyster
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ye

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tyty

pliant field
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no problem

mighty tangle
pliant field
frigid oyster