Questions in Section Linear Algebra
Find the unit vector in the direction of `v=(3, -4, 12)`.
Find the distance between the points `(8,4)` and `(0,19)`.
The angle between the vectors `u=<1,2>` and `v=<4,-1>` to the nearest tenth of a degree.
Find the reduced row echelon form of this augmented matrix: `[[13,3,-1,300],[3,1,0,100],[-6,-1,1,350]]`
Find the angle `theta` between `bbv=-3i+4j` and `bbw=5i+12j`. If necessary, round to the nearest tenth of a degree.
Consider the matrix `B=[[2,0],[0,-1]]`
- Find the characteristic polynomial of B
- Find the eigenvalues of B
- Find the associated eigenvectors of B
Solve the following system of linear equations by inverting the coefficient matrix:
`3x_1+5x_2=b_1`
`x_1+2x_2=b_2`
Write a 4x4 matrix `A=[a_(ij)]`, whose entries satisfy the following conditions:
- `a_(ij)=i^(j-1)`
- `a_(ij)={(1 if | i-j | > 1),(-1 if | i-j | <= 1):}`
Solve the following system of linear equations by Gauss-Jordan elimination method:
`x_1+x_2+2x_3=8`
`-x_1-2x_2+3x_3=1`
`3x_1-7x_2+4x_3=10`
Write the definition of a reduced row echelon form matrix. Write which of the following matrices are in the row-echelon and which are in the reduced row-echelon forms:
- `[[1,0,0],[0,0,0],[0,0,1]]`
- `[[0,1,0],[1,0,0],[0,0,0]]`
- `[[1,1,0],[0,1,0],[0,0,1]]`
- `[[1,0,2],[0,1,3],[0,0,0]]`