Step by step solution to "All bases considered in these are assumed to..."

College
Algebra
Linear algebra
Alternate coordinate systems

Efan Halliday

Answered question

2021-02-25

All bases considered in these are assumed to be ordered bases. In Exercise, compute the coordinate vector of v with respect to the giving basis S for V.

V i s P_{1}, S = t + 1, t - 2, v = t + 4

Answer & Explanation

lobeflepnoumni

Skilled2021-02-26Added 99 answers

We are given the following ordered basis S for the vector space

V = P_{1}

as well as the following vector v in V:

S = t + 1, t - 2, v = t + 4

We have to compute the coordinate vector

[v]_{S},

of v with respect to the basis S.We have

v = t + 4

= a (t + 1) + b (t - 2)

= (a + b) t + (a - 2 b) .

Equating coefficients yields the following linear system:

a + b = 1

a - 2 b = 4

The associated augmented matrix for this system is

A^{'} = [\begin{matrix} 1 & 1 & 1 \\ 1 & - 2 & 4 \end{matrix}]

Subtracting the first row from the second yields

[\begin{matrix} 1 & 1 & 1 \\ 0 & - 3 & 3 \end{matrix}]

Dividing the second row by -3 yields

[\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & - 1 \end{matrix}]

Finally, subtracting the second row from the first yields the following reduced row echelon form of

A^{'} :

A_{R}^{'} = [\begin{matrix} 1 & 0 & 2 \\ 0 & 1 & - 1 \end{matrix}]

Thus, we have the solution

a = 2

b = - 1

Therefore, the coordinate vector

[v]_{S}

of v respect to the basis S is

[v]_{S} = [\begin{matrix} 2 \\ - 1 \end{matrix}]

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