A homogeneous system with more unknowns than equations:
The Matrix Equation Ax = b and Solution Sets
The product Ax as a linear combination of A’s columns, the row-vector rule, properties of Ax, homogeneous systems and nontrivial solutions, and solution sets in parametric vector form (p + vₕ).
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All 14 slides
- 1 Slide 1
- 2 A Computation of Matrix-Vector Product x
- 3 A Computation of Matrix-Vector Product x
- 4 Example
- 5 Ax Properties of the Matrix-Vector Product
- 6 Solution Set of Linear Systems
- 7 Do you Remember
- 8 Does a nontrivial solution exist for a homogeneous
- 9 ➔ x & x are basic
- 10 Example:
- 11 Fact
- 12 Solution Set of Linear Systems
- 13 Slide 13
- 14 Slide 14
Slide 1
A Computation of Matrix-Vector Product x
Notes
Ax = x₁a₁ + … + xₙaₙ (columns weighted by x). Defined only if A has as many columns as x has entries.
A Computation of Matrix-Vector Product x
Example
Ax Properties of the Matrix-Vector Product
Notes
A(u + v) = Au + Av and A(cu) = c(Au).
Solution Set of Linear Systems
Notes
For m×n A these are equivalent: Ax = b has a solution for EVERY b ⇔ columns span ℝᵐ ⇔ A has a pivot in every ROW.
Trap Spanning ℝᵐ needs a pivot in every ROW; uniqueness needs a pivot in every COLUMN.
Do you Remember
Does a nontrivial solution exist for a homogeneous
Exam angle
➔ x & x are basic
Example:
Fact
Solution Set of Linear Systems
Notes
If Ax = b is consistent with particular solution p, every solution is p + vₕ where vₕ solves Ax = 0 (the solution set is a translated null space).
Slide 13
Slide 14
