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Yes, you when your inherit half of your home from your spouse, the basis of the inherited half increases to half of the fair market value at the death of death.

I assume you are trying to figure out the basis of your home because you sold it, and that the home wasn't used for business or rental use. If that's not right, reply with more details of your situation. 

When two people own a home and one dies, you have to consider the basis of each of the owners. The starting point is what you paid for the home, plus certain items that increase basis and others that decrease basis. 

Home improvements, but not repairs, increase the basis of your home. Follow this link for more details of adjustments to basis. 

Next, you need to figure out each owner's basis. 

For example, say your basis after you bought the home was $300,000. You spent $25,200 improving the kitchen, and $14,800 adding a bathroom. The total basis is $340,000, or $170,000 for each.

The value of the home when Spouse 2 passed was $500,000. The stepped up in basis is $250,000 for Spouse 2. 

The basis for Spouse 1 is the basis before death of $170,000, plus the basis inherited from Spouse 2 $250,000, or $420,000 total basis.