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13 Cards in this Set
- Front
- Back
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What is the parallelogram rule?
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Given vectors u and v in "r two" (in standard position), their sum u + v is the vector in standard position along the diagonal of the parallelogram determined by u and v.
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how do you perform scalar multiplication given a scalar c and a vector v?
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cv = c[v₁, v₂] = [cv₁, cv₂]
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What are the algebraic properties of vectors? How many are there?
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Let u, v, and w be vectors in "r to the n", let c and d be scalars. Then a. u + v = v + u Communtativity b. (u + v) + w = u + (v + w) Asociativity c. u + 0 = u d. u + (-u) = 0 e. c(u + v) = cu + cv Distributivity f. (c + d)u = cu + du Distributivity g. c(du) = (cd)u h. 1u = u
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What is a linear combination?
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A vector v is a linear combination of vectors v₁, v₂, ...., "v of n" if there are scalars c₁, c₂, ...., "c of n" such that v = c₁v₁, c₂v₂, ...., "c of n" times "v of n". The scalars in the linear combination are called the coeficients.
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How do you perform the dot product? Assume you are given vectors u and v.
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The dot product is performed by taking each corresponding component of u and v and multiplying them together. Each of these multiplied components is then added together. The result of this is that a dot product generates a scalar out of two vectors. u "dot product" v = v = u₁v₁ + u₂v₂ + .... + "c of n" times "v of n"
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What are the properties of the dot product? How many of them are there?
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Let u, v, and w be vectors in Rⁿ and let c be a scalar. Then a. u "dot product" v = v "dot product" u Commutativity b. u "dot product" (v + w) = u "dot product" v + u "dot product" w c. (cu) "dot product" v = c(u "dot product" v) d. u "dot product" v is greater than or equal to zero. (and u "dot product" u = 0 iff u = 0)
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What is a vector of length 1 called? What is the process of transforming a vector into a vector of length 1 called?
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A unit vector. It is called normalization.
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What is the triangle inequality
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For all vectors u and v in Rⁿ, ||u + v|| is less than or equal to ||u|| + ||v|| This is simple to realize when you consider that it basically says that the length of u + v is always greater than the length of u plus the length of v
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How do you compute the distance between two vectors u and v?
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The distance between two vectors u and v in Rⁿ is defined by d(u, v) = ||u - v||
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What does orthogonal mean and how can it be determined for two vectors?
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Two vectors u and v in Rⁿ are orthogonal to each other if u "dot product" v = 0
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Contrast Consistent vs. Inconsistent linear equations?
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A system of linear equations is called consistent if it has at least one solution. A system with no solution is called inconsistent.
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What is row echelon form?
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A matrix is in row echelon ofrm if it satisfies the following properties: 1) Any rows consisting entirely of zeros are at the bottom 2) In each nonzero row, the first nonzero entry (called the leading entry) is in a column to the left of any leading entries below it. (basically this guarantees that the leading entries form a staicase pattern)
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What elementary row operations can be performed on a matrix?
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The following elementary row operations can be performed on a matrix: 1) Interchange two rows 2) Muliply a row by a nonzero constant 3) Add a multiple of a row to another row.
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