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

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: x₁*v₁ + x₂*v₂ + ... + 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 {v₁, ..., v_p} is linearly dependent if there are weights (c₁, ..., c_p), which are not all zero, such that: c₁*v₁ + c₂*v₂* + ... + 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 v₁ is not equal to 0 then some v_j (with j > 1) is a linear combination of the preceding vectors: v₁, ..., v_j-1

• 2 Matrix Algebra

• 2.2 The Inverse of a Matrix

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• Theorem 4Let A= when is A not invertible?

  When ad-bc = 0

• How can the determinant of a 2x2 matrix A, be calculated?

  det A = ad-bc

• Theorem 6a) If A is an invertible matrix, then A-1 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.