Linear Algebra Ch. 3-4

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rainwater739
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179983
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Linear Algebra Ch. 3-4
Updated:
2012-10-25 22:51:05
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math linear algebra nul col matrix sub space vector span augment lay
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from Linear Algebra and Its Applications, 4th edition, by David C. Lay
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  1. What is the determinant of a triangular matrix?
    det A = a11 * a22 * ... * ann

    the product of the entries on the main diagonal
  2. What affect do the row operations have on the determinant?
    • For a square matrix A
    • 1. If a multiple of one row of A is added to another row to produce B, then  det B = det A
    • 2. If two rows of A are interchanged to produce B, then det B = -det A
    • 3. If one row of A is multiplied by a scalar k to produce B, then det B = k * det A
  3. Is a square matrix invertible?
    A square matrix A is invertible if and only if det A != 0
  4. Does transposing a matrix change its determinant?
    If A is a square matrix, then det AT = det A
  5. What is the multiplicative property of determinants?
    If A and B are n * n matrices, then det AB = (det A)(det B)
  6. What makes a linear system consistent?
    • A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column. In other words, the echelon form of a matrix should not have a row in the form
    • [0 ... 0 b] with b nonzero
  7. What is a span?
    If v1, ... , vp are in Rn, then the set of all linear combinations of v1, ... , vp is denoted by Span {v1, ... , vp} and is called a subset of Rn spanned by v1, ... , vp
  8. What are the properties of the matrix-vector product?
    • If A is an m * n matrix, u and v are vectors in Rn, and c is a scalar, then
    • 1. A(u + v) = A(u) + A(v)
    • 2. A(cu) = c(Au)
  9. Does a homogeneous equation have a nontrivial solution?
    The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.
  10. When is a set of vectors linearly independent?
    • An indexed set of vectors {v1 ... vp} in Rn is said to be linearly independent if the vector equation
    •                      x1v1 + x2v2 + ... + xpvp = 0
    • has the only trivial solution.
  11. When is a set of vectors linearly dependent?
    • The set {v1 ... vp} is said to be linearly dependent if there exists weights c1, c2, ... , cp, not all zero, such that
    •                c1v1 + c2v2 + ... + cpvp = 0
  12. How can you tell if two vectors are linearly dependent?
    A set of two vectors is linearly independent if at least one of the vectors is a multiple of the other.
  13. If a set contains more vectors than there are entries in each vector, is the set linearly dependent or independent?
    If a set contains more vectors than there are entries in each vector, the set would be linearly dependent.
  14. If a set contains the zero vector, is the set linearly dependent or independent?
    If a set contains the zero vector, the set would be linearly dependent.
  15. What is a transformation?
    A transformation (or function, or mapping) T from Rn to Rm is a rule that assigns te each vector x in Rn a vector T(x) in Rm, where the set Rn is the domain and the set Rm is the range.
  16. How do you know if a transformation is linear?
    • A transformation T is linear if, for all vectors u, v and all scalars c
    • 1. T(u + v) = T(u) + T(v)
    • 2. T(cu) = cT(u)
  17. What are the properties of matrix addition and multiplication by scalars?
    • Let A, B, and C be matrices of the same size, and let r and s be scalars
    • 1. A + B = B + A
    • 2. (A + B) + C = A + (B + C)
    • 3. A + 0 = A
    • 4. r(A + B) = rA + rB
    • 5. (r + s)A = rA + sA
    • 6. r(sA) = (rs)A
  18. What are the properties of matrix multiplication?
    • Let A be an m * n matrix, and let B and C have sizes for which the indicated sums and products are defined, and let r be any scalar
    • 1. A(BC) = (AB)C, associative law of multiplication
    • 2. A(B + C) = AB + AC, left distributive law
    • 3. (B + C)A = BA + CA, right distributive law
    • 4. r(AB) = (rA)B = A(rB)
    • 5. ImA = A = AIn
  19. What are the properties of transposed matrices?
    • Let A and B denote matrices whose sizes are appropriate for the following sums and products, and let r be any scalar
    • 1. (AT)T = A
    • 2. (A + B)T = AT + BT
    • 3. (rA)T = rAT
    • 4. (AB)T = BTAT
  20. What are the equivalent statements in the Invertible Matrix Theorem?
    • Let A be a square n * n matrix, then the given statements are either all true or all false
    • 1. A is an invertible matrix
    • 2. A is row equivalent to the n * n identity matrix
    • 3. A has n pivots
    • 4. The equation Ax = 0 has the only trivial solution
    • 5. The columns of A form a linearly independent set
    • 6. The linear transformation x |-> Ax is one-to-one
    • 7. The equation Ax = b has at least one solution for each b in Rn
    • 8. The columns of A span Rn
    • 9. The linear transformation x |-> Ax maps Rn onto Rn
    • 10. There is an n * n matrix C such that CA = I
    • 11. There is an n * n matrix D such that DA = I
    • 12. AT is an invertible matrix

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