Term
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Definition
Any collection of S vectors in R^n such that:
1. The zero vector is in S
2. If u and v are in S, then u+v is in S.
3. If u is in S and c is a scalar, then cu is in S. |
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Term
| A basis for a subspace S of R^n |
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Definition
A set of vectors in S that:
1. Spans S and
2. is linearly dependent |
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Term
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Definition
| If S is a subspace of R^n, then the number of vectors in a basis for S is called the dimension of S, denoted dim S. |
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Definition
| The dimension of its row and column spaces and is denoted by rank(a) |
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Definition
| The dimension of its null space and is denoted by nullity(a) |
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Definition
A trasnformation T: R^n->R^n is called a linear transformation if:
1. T(u+v)=T(u)+T(v) for all u and v in R^n and
2. T(cv)=cT(v) for all v in R^n and all scalars C |
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Term
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Definition
| Let A be an nxn matrix. A scalar lambda is called an eigenvalue of A if there is a nonzero vector x such that Ax=lambdax. Such a vector x is called an eigenvector of A corresponding to lambda. |
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Term
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Definition
| Let A be an nxn matrix and let lambda be an eigenvalue of A. The collection of all eigenvectors corresponding to lambda together with the zero vector is called the eigenspace of lambda and is denoted by Esublambda. |
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Definition
| If A is an mxn matrix, then: rank(A)+nullity(A)=n |
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