Linear Algebra Ch. 3-4

det A = a₁₁ * a₂₂ * ... * a_nn

the product of the entries on the main diagonal

• 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

A square matrix A is invertible if and only if det A != 0

If A is a square matrix, then det A^T = det A

If A and B are n * n matrices, then det AB = (det A)(det B)

• 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

If v₁, ... , v_p are in Rⁿ, then the set of all linear combinations of v₁, ... , v_p is denoted by Span {v₁, ... , v_p} and is called a subset of Rⁿ spanned by v₁, ... , v_p

• 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)

The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.

• An indexed set of vectors {v₁ ... v_p} in Rⁿ is said to be linearly independent if the vector equation
•                      x₁v₁ + x₂v₂ + ... + x_pv_p = 0
• has the only trivial solution.

• The set {v₁ ... v_p} is said to be linearly dependent if there exists weights c₁, c₂, ... , c_p, not all zero, such that
•                c₁v₁ + c₂v₂ + ... + c_pv_p = 0

A set of two vectors is linearly independent if at least one of the vectors is a multiple of the other.

If a set contains more vectors than there are entries in each vector, the set would be linearly dependent.

If a set contains the zero vector, the set would be linearly dependent.

A transformation (or function, or mapping) T from Rⁿ to R^m is a rule that assigns te each vector x in Rⁿ a vector T(x) in R^m, where the set Rⁿ is the domain and the set R^m is the range.

• 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)

• 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

• 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. I_mA = A = AI_n

• Let A and B denote matrices whose sizes are appropriate for the following sums and products, and let r be any scalar
• 1. (A^T)^T = A
• 2. (A + B)^T = A^T + B^T
• 3. (rA)^T = rA^T
• ^4. (AB)^T = B^TA^T

• 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. A^T is an invertible matrix