Summary: Linear Algebra And Its Applications  9780134013473  David C Lay, et al
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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 Linear Equations in Linear Algebra

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: x_{1}*v_{1} + x_{2}*v_{2} + ... + x_{p}*v_{p} = 0, has only the trivial solution. 
When is an indexed set of vectors {v1, ..., vp} in Rn said to be linearly dependent?
The set {v_{1}, ..., v_{p}} is linearly dependent if there are weights (c_{1}, ..., c_{p}), which are not all zero, such that: c_{1}*v_{1} + c_{2}*v_{2}* + ... + c_{p}*v_{p} = 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 v_{1} is not equal to 0 then some v_{j} (with j > 1) is a linear combination of the preceding vectors: v_{1}, ..., v_{j1} 
2 Matrix Algebra

2.2 The Inverse of a Matrix
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Theorem 4Let A= when is A not invertible?
When adbc = 0 
How can the determinant of a 2x2 matrix A, be calculated?
det A = adbc 
Theorem 6a) If A is an invertible matrix, then A1 is invertible and?
(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:
(A^{T})^{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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