The United States today said it would give $15 million in economic aid to Rhodesia, which will become the independent state of Zimbabwe Friday.

The development reflects what U.S. officials describe as the Carter administration's determination to act "quickly and conclusively" in showing support for Rhodesian Prime Minister Robert Mugabe, despite the fact that he is an avowed Marxist.

According to these officials, the aid package is intended to "demonstrate to him that Zimbabwe's future lies with the West."

The $15 million in aid, which already has received congressional approval, is earmarked for reconstruction and rehabilitation in rural areas, where an estimated 1 million people were displaced by fighting.

The officials said Rhodesia was in urgent need of health clinics and improved transportation and communications.

They said the American aid would be part of an overall package of assistance now being put together by Britain and which is expected to total more than $40 million. Britain will contribute about $15 million and other West European countries were to make up the remainder.

American officials privately suggest that the United States should increase its aid to the Mugabe government to "at least $40 million this year." About $20 million in aid has been approved so far.

Administration officials say that Washington's objectives are twofold in extending aid to Rhodesia. First, the peaceful evolution of biracial democracy in Zimbabwe would provide an encouraging precedent of peaceful transition for Namibia and South Africa. Second, the successful development of Zimbabwe could tempt nearby Angola and Mozambique to loosen ties with Moscow.

U.S. officials believe that if the West fails to come to Mugabe's aid or displays hostility to his government, he would not hesitate to seek Soviet assistance. The Soviets supported Joshua Nkomo's guerrilla faction during the seven-year civil war and Mugabe has no ties yet with Moscow.

In Salisbury, meanwhile, British governor Lord Soames praised Mugabe for the conciliatory stance he has taken since winning Rhodesia's independence elections seven weeks ago.

"I have noted with admiration and respect the spirit of reconciliation which has formed the prime minister's word and actions over these weeks and his obvious determination to sustain the unity of the nation and its economic strength," Soames said.