Question

g Are the following statements true or false? 1. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.

Answer 1

Answer:

The statement is true that any linear combination of vectors can always be written in the form Ax for suitable matrix A and vector x.

Step-by-step explanation:

Let

a1x1 + a2x2 + a3x3 + ... + anxn = d1

b1x1 + b2x2 + b3x3 + ... + bnxn = d2

.

.

.

c1x1 + c2x2 + c3x3 + ... + cnxn = dn

be a linear system, and let

[a1 a2 a3 ... an][x1] = [d1]

[b1 b2 b3 ... bn][x2] [d2]

[. . . .

[. . . .

[. . . .

[c1 c2 c3 ... cn][xn] [dn]

be the augmented matrix of the linear system. The augmented matrix can be represented as Ax = d

Where A is the matrix of (a1, a2,...., an), x is (x1, x2, x3, ..., xn), and d = (d1, d2, d3,..., dn) .

The solution set to the linear system is the same as the solution set of Ax = d.