Question

write the matrix A[3,-1,1,-2]as a linear combination of A1=[1,1,0,-1],A2=[1,1,-1,0]and A3=[1,-1,0,0]

Answer 1

Write the matrix $A = [3, -1, 1, -2]$ as a linear combination of $A1 = [1, 1, 0, -1]$, $A2 = [1, 1, -1, 0]$ and $A3 = [1, -1, 0, 0]$

**Solution:**

We need to present the matrix $A$ in the form:

$$A = x * A_1 + y * A_2 + z * A_3$$

where $x, y, z$ some constants.

$$\begin{array}{l} x * A_1 + y * A_2 + z * A_3 = x * [1, 1, 0, -1] + y * [1, 1, -1, 0] + z * [1, -1, 0, 0] \\ = [x, x, 0, -x] + [y, y, -y, 0] + [z, -z, 0, 0] \\ = [x + y + z, x + y - z, 0 - y + 0, -x + 0 + 0] \\ = [x + y + z, x + y - z, -y, -x] \end{array}$$

So we have that

$$[x + y + z, x + y - z, -y, -x] = A$$ $$[x + y + z, x + y - z, -y, -x] = [3, -1, 1, -2]$$

We have next system of linear equation:

$$\left\{ \begin{array}{l} x + y + z = 3 \\ x + y - z = -1 \\ -y = 1 \\ -x = -2 \end{array} \right.$$ $$\left\{ \begin{array}{l} 2z = 4 \\ y = -1 \\ x = 2 \end{array} \right.$$ $$\left\{ \begin{array}{l} x = 2 \\ y = -1 \\ z = 2 \end{array} \right.$$

So matrix $A$ can be present as

$$A = 2A_1 - A_2 + 2A_3$$

**Answer:** $A = 2A_1 - A_2 + 2A_3$