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15 Cards in this Set
- Front
- Back
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Theorem 4 |
1. For each b in Rm Ax = b has a solution 2. Each b in Rm is a linear combination of columns A 3. Columns of A span Rm 4. A has a pivot position in every Row |
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Theorem 5 |
A(u + v) = Au + Av A(cu) = c(Au) |
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Homogeneous |
If a system can be written in form Ax = 0 |
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Trival solution |
A Homogeneous equation with a 0 vector solution |
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Non trivial solution |
A homogeneous equation with more then 1 solution (I.e Ax = 0 multipull ways to get 0). |
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Theorem 6 |
Ax = b is consistent for ome b. Ap = b; Then the solution set of Ax = b is the set of all vectors of form w = p + vh. (vh is any solution of Ax= b (i.e an x). |
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Linear independent |
No free variables |
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Linear dependant |
Free variables |
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Theorem 7 |
If a vector is outside the span of a matrix, then the matrix is linear independant. |
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Theorem 8 |
If there are more columns then rows, then a solution set is linear dependant. |
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Theorem 9 |
If a set contains a zero vector, then it is Linearly dependent. |
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When is a transformation linear? |
If T(u + v) = T(u) + T(v) if T(cu) = cT(u) |
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Theorem 10 |
T : Rn -> Rm There exists a unique matrix A such that T(x) = Ax for all x in Rn A is the mxn matrix whose jth vector is T(ej) where ej is the jth column of the identity matrix in Rn. |
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Theorem 11 |
T : Rn -> Rm T is one to one if and only if the equation T(x) = 0 has only the trivial solution |
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Theorem 12 |
T : Rn - > Rm is a linear transformation. A standard matrix. A. T maps Rn onto Rm if and only if the columns of A span Rm. B. T is one to one if and only if the columns of A are linearly independent. |
