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Definition
The linear combination of the columns of A using the corresponding entries in x as weights
a1x1 + a2x2 + … + anxn |
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| The set of all scalar multiples of v. Visualized as the set of points on the line in R3 and through v and 0. Geometric interpretation in R3 is a line through the origin. |
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| If u and v are nonzero vectors in R3, with v not a multiple of u, the Span {u,v} is the plane in R3 that contains u, v and 0. In particular, Span {u,v} contains the line in R3 through v and 0. Geometric interpretation in R3 is a plane through the origin. |
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Definition
If v1, . . . ,vp are in Rn, then the set of all linear combinations of v1, . . . ,vp is denoted by Span {v1 . . . vp} and is called the subset of Rn spanned (or generated) by v1, . . . ,vp. That is, Span {v1 . . . vp} is the collection of all vectors that can be written in the form
c1v1 + c2v2 + . . . + cpvp
with c1, . . . , cp scalars. |
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An indexed set of vectors {v1 . . . vp} in Rn is said to be linearly independent if the vector equation:
x1v1 + x2v2 + . . . + xpvp = 0
has only the trivial solution. |
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The set of vectors {v1 . . . vp} is said to be linearly dependent if there exists weights c1, . . . , cp, not all zero, such that:
c1v1 + c2v2 + . . . + cpvp = 0. |
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Definition
A transformation (or mapping) T is linear if:
(i) T(u + v) = T(u) + T(v) for all u, v in the domain of T
(ii) T(cu) = cT(u) for all u and scalars c |
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| Standard matrix (for a linear transformation T) |
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| The matrix A such that T(x) = Ax in the domain of T. |
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Definition
If A is an m x n matrix, and if B is an n x p matrix with columns b1, . . . , bp, then the product AB is the m x p matrix whose columns are Ab1, . . . , Abp. That is
AB = A[b1 b2 … bp] = [Ab1 Ab2 … Abp] |
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| A mapping T: Rn Rm such that each b in Rm is the image of at most one x in Rn. |
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| A mapping T: Rn Rm such that each b in Rm is the image of at least one x in Rn. |
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Definition
| The representation of a matrix A in the form A = LU where L is a square lower triangular matrix with ones on the diagonal (a unit lower triangular matrix) and U is an echelon form of A. |
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