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theRukus
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Homework Statement
Let {X, Y, Z} be linearly independent in Rn. If {X, Y, Z, W} is linearly dependent, show that W [tex]\epsilon[/tex] span{X, Y, Z}. NB: You must SHOW this.
Homework Equations
The Attempt at a Solution
For W to belong to the span of {X,Y,Z}, W = aX + bY + cZ where a, b, c are some real numbers. Since {X,Y,Z,W} is linearly dependent, one of the four vectors can be shown as a linear combination of the other three, ie. W = aX + bY + cZ, where a, b, c are some real numbers.
On my assignment, I have the previous statement, and I've also represented X, Y, Z, and W with {x1, x2, x..., xn}, etc. I've taken these equations and multiplied the real numbers in.
I know what I'm trying to say here, but I don't exactly know how to say it, if you know what I mean. Is my explanation enough? Does it SHOW that W belongs to the span of {X, Y, Z}?
Thanks,
Connor Bode