# Summary: Linear Algebra And Its Applications | 9780134013473 | David C Lay, et al • This + 400k other summaries
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## Read the summary and the most important questions on Linear Algebra and Its Applications | 9780134013473 | David C. Lay; Steven R. Lay; Judi J. McDonald

• ### 1.2 Row Reduction and Echelon Forms

• #### A rectangular matrix is in echelon form (or row echelon form) if it has thefollowing three properties:

1. All nonzero rows are above any rows of all zeros.
2. Each leading entry of a row is in a column to the right of the leading entry of
the row above it.
3. All entries in a column below a leading entry are zeros.
• #### If a matrix in echelon form satisfies the following additional conditions, then it is in reduced echelon form (or reduced row echelon form):

4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only nonzero entry in its column.
• ### 1.7 Linear Independence

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• #### When is an indexed set of vectors {v1, ..., vp} in Rn said to be linearly independent?

When the vector equation: x1*v1 + x2*v2 + ... + xp*vp = 0, has only the trivial solution.
• #### When is an indexed set of vectors {v1, ..., vp} in Rn said to be linearly dependent?

The set {v1, ..., vp} is linearly dependent if there are weights (c1, ..., cp), which are not all zero, such that: c1*v1 + c2*v2* + ... + cp*vp = 0
• #### Theorem 7: Characterization of Linearly Dependent Sets An indexed set S = {v1, ..., vp} of two ore more vectors is linearly dependent if and only if?

At least one of the vectors in S is a linear combination of the others. Furthermore, if S is linearly dependent and v1 is not equal to 0 then some vj (with j > 1) is a linear combination of the preceding vectors: v1, ..., vj-1

• ### 2.2 The Inverse of a Matrix

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(A-1)A-1 = A
• #### Theorem 6c) If A is an invertible matrix, then so is AT, what is the inverse of AT

The transpose of A-1, Giving:

(AT)-1 = (A-1)T
• #### Theorem 6b generalizationThe product of n x n invertible matrices is?

Invertible, and the inverse is the product of the inverses in the reverse order.
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