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| Let V be a subspace of R^n for some n. A collection B = {v1, v2, …} of vectors from V is said to be a basis for V if _____ |
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| B is linearly independent and spans V |
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| How do you verify that nonempty subset W of vector space V is a subspace of V? |
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| Show closure of u_+c_*_v in W for arb vectors u, v and arb scalar c. |
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| What true about span of a set of vectors from vector space V? |
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| What can you say of about Span of a single nonzero vector in R2 or R3? |
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| The set is a subspace. Graphically, it's a line that passes thru origin. |
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| What can you say of about Span of two linearly indi nonzero vectors in R2 or R3? |
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| The set is a subspace. Graphically, it's a plane that passes thru origin. |
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| a subset of a vector space that is closed under addition and scalar multiplication |
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| Intersection of subspaces a subspace? How about the union of two subspaces? |
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| Intersection is a subspace. Union might be. |
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| The linear system Ax=b is consistent iff ____ |
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| b is in the column space of A |
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| If A is mXn matrix, null(A) is subspace of ___ and col(A) is subspace of ___ |
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| null(A) is subspace of R^n and col(A) is subspace of R^m |
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| Every nontrivial vector space has ___ many bases. |
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| dim(R^n) = __ ; dim(M_{mXn}) = __ ; dim(P_n) = ; __ |
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| dim(R^n) = n ; dim(M_{mXn}) = mn ; dim(P_n) = ; n+1 |
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| If the span of a set B of n vectors is V and dim(V)=n then |
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| If B and B' are two ordered bases for V, the transition matrix from B to B' is __ and the inverse matrix is __ |
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| transition matrix from B to B' is invertible and invertible matrix is transition matrix from B to B' |
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| Given any two ordered bases for vector space, V, a a transition matrix can be used to |
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| change the coords of a vector relative to one basis to the coords relative to other basis |
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| T is fn from vector spaces V into W. T is linear transformation provided that __ |
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| for all u,v in V and all scalars c, T(cu+v) = cT(u)+T(v) |
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| If A is mXn matrix and T(x)=Ax then T is __ |
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| linear transformation from R^n into R^m |
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| If T is a linear transformation, then T(0)= |
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| linear combinations of linear transformations are |
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| If T:V-->W and L:W-->Z are both linear transformations, then |
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| LoT:V-->Z is a linear transformation |
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| let W be a subset of vector space V. W is subspace iff following 3 conditions hold: |
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| zero vector, 0, is in W; Closure of any linear combo (scalar mult too) in W |
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