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De Moivre's Theorem : Write a cis expression in standard complex form (a+bi)
Find the real and complex solution of the equation Xsquared - 8 = 0
Find the angle between the two vectors (5,2) and (-2,5)
Use Demoivre's theorem to write [3cis(pie/2)]squared in the form a+bi without trigonometric functions.
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Vectors and Steinitz replacement
Illustrate the steps of the Steinitz replacement theorem by converting B into A step-by-step. Please see the attachment.
We consider the following two set of vectors.
, , where
, , , , ,
(a) I claim that both and are basis in .
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Linear Algebra - Four Fundamental Subspaces
We note that is actually a linear combination of the column vectors of . Here is a simple explanation.
,
So is a linear combination of the column vectors of , thus .
From Theorem 4.7, we have , thus we have
Thus .
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Vector Spaces and Subspaces
Use Theorem 5.2.1 to determine which of the following are subspaces of R3.
Thm 5.2.1: If W is a set of one or more vectors from a vector space V, then W is a subspace of V if and only if the following conditions hold.
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Vector Space : Linear Dependence and Null Space
The vectors spans a subspace of dimension and . Then any vectors with among the vectors are linearly dependent. So the number of linear dependencies is: .
For any , we can find , such that
.
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Vector Subspaces
If you get a row of zeroes, then you have discovered that a linear combination of at least two of the vectors can produce another member of the spanning set. Therefore the row vectors do NOT span Rn (i.e. not a basis).
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Pythagorean Theorem
Show that |v - u|^2 = |u|^2 + |v|^2 under these conditions, and explain why this is the Pythagorean theorem. The vectors u and v are perpendicular in this situation, so the dot product u.v = 0.
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Prove the Extended Pythagorean Theorem: E^2=lbl
E = | b - p |, where E, b and p are all vectors.
Squaring both sides,
E2 = | b - p |2
= (b - p).
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Linear Algebra: Basis and Dimension
However, I think we only need to use one theorem.
Assume that we have a finite-dimensional vector space V, and W is an infinite dimensional subspace of V.
By theorem 5.4.7, dim(W) ≤ dim(V).