Two different ways to express begin{bmatrix}11 end{bmatrix} as a linear combination of v_{1}, c_{2} and v_{3}.

Question

Two different ways to express

[\begin{matrix} 1 \\ 1 \end{matrix}]

as a linear combination of

v_{1}, c_{2} a n d v_{3} .

Answer 1

Given vectors are

v_{1} = [\begin{matrix} 1 \\ - 3 \end{matrix}], v_{2} = [\begin{matrix} 2 \\ - 8 \end{matrix}] a n d v_{3} = [\begin{matrix} - 3 \\ 7 \end{matrix}]

Calculation:
Let the vector w =

[\begin{matrix} 1 \\ 1 \end{matrix}]

.
The vector w can be represented in terms of as

w = a v_{1} + v v_{2} + c v_{3} .

Determine the coefficients a, b, c.
Write the vectors

v_{1}, v_{2} a n d v_{3}

as the columns of a matrix

[v_{1} v_{2} v_{3} 2]

and reduce the matrix in to row reduce echelon form.

[\begin{matrix} 1 & 2 & - 3 & 1 \\ - 3 & - 8 & 7 & 1 \end{matrix}] R_{2} \to 3 R_{1} + R_{2} [\begin{matrix} 1 & 2 & - 3 & 1 \\ 0 & - 2 & - 2 & 4 \end{matrix}]

[\begin{matrix} 1 & 2 & - 3 & 1 \\ 0 & - 2 & - 2 & 4 \end{matrix}] R_{2} \to - \frac{1}{2} R_{2} [\begin{matrix} 1 & 2 & - 3 & 1 \\ 0 & 1 & 1 & - 2 \end{matrix}]

[\begin{matrix} 1 & 2 & - 3 & 1 \\ 0 & 1 & 1 & - 2 \end{matrix}] R_{1} \to R_{1} - 2 R_{2} [\begin{matrix} 1 & 0 & - 5 & 5 \\ 0 & 1 & 1 & 2 \end{matrix}]

This gives the following equations,

a - 5 c = 5

b + c = - 2

Here, c is a free variable.
Two find two different ways to write w as a linear combination of

v_{1}, v_{2}, v_{3},

choose two different values of c, which will give two different values of corresponding a and b.
Let,

c = 1

, this gives,

a = 10

b = - 3

Then, the vector w can be expressed as

w = 10 v_{1} - 3 v_{2} + v_{3} .

Let

c = 2,

this gives,

a = 15

b = - 4

Then, the vector w can be expressed as

w = 15 v_{1} - 4 v_{2} + 2 v_{3} .

Therefore, two different ways of expressing

[\begin{matrix} 1 \\ 1 \end{matrix}]

as a linear combination of

v_{1}, v_{2}, v_{3} a r e 10 v_{1} - 3 v_{2} + v_{3} a n d 15 v_{1} - 4 v_{2} + 2 v_{3} .

Expert Answer