Question

The set

B={[1. 1] , [0 1] , [0 0]}
{[0. 0] , [0 -1] , [0 -2]}

is a basis of the space of upper-triangular 2×2 matrices.

find the coordinates of
m= [−2 −3] with respect to this basis.
[0 -5]

Answer 1

To find the coordinates of the matrix M with respect to the basis B, we express M as a linear combination of the matrices in B. Solving for the coefficients, we find that the coordinates of M are (-2, -1, 0, 0, -5, 0).

Step-by-step explanation:

Finding the Coordinates of a Matrix With Respect to a Basis

The student has asked to find the coordinates of the matrix M = [−2 −3]

[0 −5] with respect to the basis B = { [1 1], [0 1], [0 0] }, { [0 0], [0 -1], [0 -2]} in the space of upper-triangular 2×2 matrices. To find these coordinates, we need to express M as a linear combination of the matrices in B.

We want coefficients a, b, c such that:

M = a[1 1] + b[0 1] + c[0 0] + d[0 0] + e[0 -1] + f[0 -2]

Equating M with the right-hand side matrix gives us:

• −2 = a;
• −3 = a + b;
• 0 = c; (since M is upper-triangular, the bottom-left element is always 0)
• 0 = d;
• −5 = e;
• 0 = f.

Solving this system, we find that a = −2, b = −1, c = 0, d = 0, e = −5, and f = 0. Therefore, the coordinates of M with respect to B are (−2, −1, 0, 0, −5, 0).