Let v be the vector space of all 2 x 2 matrices over the field f. prove that v has dimension 4 by exhibiting a basis for v which has four elements.

See the proof below.

Step-by-step explanation:

We can define a basis of V with the following elements:

So then if we define the basis X as following:

We see the the dimension for X is 4 since the basis have a dimension of 4

or

step-by-step explanation:

we need to solve for 'x' the following equation:  

we have that:

⇒

⇒ ⇒

factorizing, we get that:

  ⇒

then, the solutions are: or

nothing extraneous has been found!

"standard form:

3x2 + 14x − 10 = 0

solutions based on quadratic formula:

x1   = −14 − √ 142 − 4×3×(−10) /   2×3   = −14 − 2 × √ 79 /   6   ≈ −5.296

x2   = −14 + √ 142 − 4×3×(−10) /   2×3   = −14 + 2 × √ 79 /   6   ≈ 0.629