Question

: In Exercises 1-4, W is a subspace of the vector space V of all (2 x 2) matrices. A matrix A in W is written as a b -=[:] In each case exhibit a basis for W. 4. W = {A: b = a - c, d = 2a + c)

Answer 1

To find a basis for W, we need to find a set of linearly independent matrices in W that span W.

Let's first rewrite the condition for A to be in W:

A = [a a - c

b 2a + c]

We can rewrite A in terms of a linear combination of matrices as follows:

A = a [1 1

0 2] + b [0 -1

1 0] + c [0 1

0 1]

Therefore, any matrix A in W can be written as a linear combination of the three matrices:

B1 = [1 1

0 2],

B2 = [0 -1

1 0],

B3 = [0 1

0 1]

We just need to check that these three matrices are linearly independent. To do this, we set up the equation

c1 B1 + c2 B2 + c3 B3 = 0

where c1, c2, c3 are scalars. This gives the system of linear equations

c1 = 0

c2 - c3 = 0

c1 + c2 + c3 = 0

The solution to this system is c1 = 0, c2 = c3, and any value for c2. This means that the three matrices are linearly independent, and hence they form a basis for W.

give thanks for more! welcome <3

Explanation: