411 REFERENCE · Wikibooks
Linear Algebra and the C Language/a0ms
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Matrix of a linear application with respect to the basis B:
Let T(x_s) = A x_s a linear application with respect to the standard basis.
Let [T(x_b)]_b = D x_b the linear application with respect to the basis B.
Study the relationship between the matrix A and the matrix D.
We have:
T(x_s) = A x_s In standard basis
[T(x_s)]_b = [A x_s]_b In the B basis
We know that: [U]_b = InvB [U]_s then [A x_s]_b = InvB [A x_s]_s = InvB A x_s
[T(x_s)]_b = InvB (A x_s)
[T(x_s)]_b = (InvB A)(x_s)
We know that: U_s = B U_b then (x_s) = (B x_b)
[T(x_b)]_b = (InvB A)(B x_b)
[T(x_b)]_b = (InvB A B) x_b
D = (InvB A B) is the relationship between A and D
...a.c : Calculate vectors x_b and x_s
...b.c : Calculate T(x) = A x_b
...c.c : Calculate [T(x_b)]_b = D x_b
...d.c : Check if: B T(x_b)_b = T(x_s)
Example in R2:
+ * c02a.c * c02b.c * c02c.c * c02d.c
Example in R3:
+ * c03a.c * c03b.c * c03c.c * c03d.c
Example in R4:
+ * c04a.c * c04b.c * c04c.c * c04d.c