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Definition
a nonempty set V of vectors on which are defined two operations: addition and multiplication by scalars. the following must be true:
- the sum of u and v are in V
- the scalar multiple of u, denoted as cu, must be in V
- the zero vector must be included in V
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| spanning set of a subspace H |
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Definition
| if v1,...,vp are in a subspace H, then Span{v1,...,vp} is a subspace of H |
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| linear transformation from one vector space to another |
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Definition
a linear transformation T from a vector space V into a vector space W is a rule that assigns to each vector x in V a unique vector T(x) in W, such that
- T(u+v) = T(u) + T(v)
- T(cu) = cT(u)
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Definition
| {v1,...,vp} in ℝn is said to be linearly independent if the vector equation x1v1 + x2v2 + ... + xpvp = 0 has only the trivial solution. the set {v1,...,vp} is said to be linearly dependent if there exist weights c1,...,cp, not all zero, such that c1v1 + c2v2 + ... + cpvp = 0. |
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Definition
a subpsace of ℝn is any set H in ℝn that has three properties:
- the zero vector is in H
- for each u and v in H, u+v is in H
- for each u in H and each scalar c, the vector cu is in H
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Definition
| a basis for subspace H of ℝn is a linearly independent set in H that spans H |
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Definition
| the dimension of a nonzero subspace H, denoted by dimH, is the number of vectors in any basis for H. the dimension of the zero subspace is defined to be zero. |
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Definition
| a mapping T : ℝn → ℝm is said to be one-to-one if each b in ℝm is the image of at most one x in ℝn |
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Definition
| a mapping T : ℝn → ℝm is said to be onto ℝm if each b in ℝm is the image of at least one x in ℝn |
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Definition
| if the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set |
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Definition
| if n x n matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities) |
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Definition
| the set RowA of all linear combinations of the vectors formed from the rows of A; also denoted as ColAT if two matrices A and B are row equivalent, then their row spaces are the same. |
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Definition
| the column space of an m x n matrix A, written as ColA, is the set of all linear combinations of the columns of A. if A = [a1,...an], then ColA = Span{a1,...an} |
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Definition
| the null space of an m x n matrix A, written as NulA, is the set of all solutions of the homogeneous equation Ax = 0. in set notation, NulA = {x : x is in ℝn and Ax = 0} |
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Term
span{v1,...,vp}
subspace spanned by v1,...,vp |
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Definition
| the set of all linear combinations of v1,...,vp |
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