Assignment #3: Matrix Properties and Operations (back to full lecture notes)
In this assignment you will perform step-by-step algebraic manipulations involving vector and matrix properties and operations.
You may not add new lines or premises above an "implies" logical operator. If you add any
new premises, you will recieve no credit.
- Finish the argument below that shows that matrix multiplication of vectors in R3 preserves lines.
- Finish the argument below that shows that if multiplication by a matrix M maps three vectors to [0; 0; 0], it
maps any linear combination v' of those vectors to [0; 0; 0].
Extra credit: if u, v, and w are setwise linearly independent, what can you say about the matrix M? Be as specific as
- Show that if a matrix M in R2×2 is invertible, there is a unique solution for u given any equation M u = v with v ∈ R2.
- Show that if the column vectors of a matrix in R2×2 are linearly dependent, then its determinant is 0.
Extra credit: you may include the opposite direction of this proof for extra credit (build a new, separate argument in which
det [a,c; c,d] = 0 is found above the
`([a;c]) and ([b;d]) are linearly dependent`
is at the end of the argument below it).
- In this problem, you will define explicit matrices that correspond to elementary row operations on matrices in R2×2.
- Find appropriate matrices A, B, C, and D in R2×2 to finish the following argument.
- Use the matrices A, B, C, and D from part (a) with matrix multiplication to construct a matrix E that can be
shown to satisfy the last line in the following argument.
- Extra credit: the row operations defined by A, B, C, and D are all invertible as long as t ≠ 0;
prove this for t = -1 by showing that the matrices A, B, C, and D are invertible.