Questions in Section Linear Algebra

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:

1. `a_(ij)=i^(j-1)`
2. `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. `[[1,0,0],[0,0,0],[0,0,1]]`
2. `[[0,1,0],[1,0,0],[0,0,0]]`
3. `[[1,1,0],[0,1,0],[0,0,1]]`
4. `[[1,0,2],[0,1,3],[0,0,0]]`

Consider the following system of linear equations:

`x + y + 2z = a`

`x + z = b`

`2x + y + 3z = c`

Show that for this system to be consistent, the constants must satisfy the condition: `c=a+b`.

Find the augmented matrix of the following system of linear equations:

`2x_1+2x_3=1`

`3x_1-x_2+4x_3=7`

`6x_1+x_2-x_3 =0`

Find inverse matrix `A^(-1)` by Gaussian elimination: `A=([1,2,1],[1,3,1],[0,2,1])`.

Write the following matrix system in matrix form. Find `A^(-1)` using Gauss-Jordan elimination. Use `A^(-1)` to then solve the system. All calculations should be exact (use fractions).

`-5x+y+4z = 10`.

Given vectors `v=(1,2,3)` and `w=(1,-1,5)`.

1. Calculate the angle between `v` and `w` correct to 2 decimal places.
2. Find two unit vectors orthogonal to `v` and `w`.
3. Find the vector component of `v` along `w` and the vector component of `v` orthogonal to `w`.

Let `v=[[v_1],[v_2]]`. Describe the set `H` of vectors `[[x_1],[x_2]]` that are orthogonal to `v`.