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Definition
| The linear combination of the columns of A using the corresponding entries in x as weights: x1a1 + x2a2 + ... + xnan |
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| Define Span{v} & its geometric interpretation |
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Definition
| The set of all scalar multiples of v. Geometrically, it is visualized as a set of points on a line in R^n |
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| Define Span{u,v} & its geometric interpretation |
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Definition
| 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 R^n, then the set of all linear combinations of v1,...,vp is denoted by Span {v1,...,vp} and is called the subset of R^n 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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| Define Linear Independence |
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Definition
An indexed set of vectors {v1,...,vp} in R^n is said to be linearly independent if the vector equation:
x1v1 + x2v2 + . . . + xpvp = 0
has only the trivial solution. |
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Definition
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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| Define Linear Transformation |
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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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| Define a Standard matrix of a linear transformation |
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Definition
| The matrix A such that T(x) = Ax in the domain of T. A = [T(e1)...T(en)] |
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Term
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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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| Define One to one transformation |
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Definition
| A mapping T:R^n->R^m such that each b in R^m is the image of at most one x in R^n. |
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| Define Onto transformation |
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Definition
| A mapping T:Rn->Rm such that each b in R^m is the image of at least one x in R^n. |
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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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| When are {v1,v2} linearly dependent |
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Definition
| If one of the vectors is a multiple of the other. |
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| Describe AB using the columns of AB |
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Definition
| Each column of AB is a linear combination of the columns of A using weights from the correspoind column of B. |
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| For AB, what does row-i(AB)= ? |
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Definition
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| What does the transpose of a product of matrices equal? |
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Definition
| The transpose of a product of matrices equals the product of their transposes in the reverse order. |
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Definition
A subspace of R^n is any set H in R^n that has three properties:
a. The zero vector is in H
b. For each u and v in H, the sum u+v in H
c. For each u in H and each scalar c, the vector cu is in H |
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| Define the column space of a matrix |
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Definition
| The column space of a matrix A is the set Col A of all linear combinations of the columns of A |
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| Define the null space of a matrix |
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Definition
| The null space of a matrix A is the set Nul A of all solutions to the homogeneous equation Ax = 0. |
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| Define a basis for a subspace |
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Definition
| A basis for a subspace H of R^n is a linearly independent set in H that spans H |
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