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In Hindu tradition, Saraswati has retained her significance as a goddess from the Vedic age up to the present day. In Shanti Parva of the Hindu epic Mahabharata, Saraswati is called the mother of the Vedas, and later as the celestial creative symphony who appeared when Brahma created the universe. In Book  Taittiriya Brahmana, she is called “the mother of eloquent speech and melodious music”. Saraswati is the active energy and power of Brahma.

Who is the Guru of Goddess Saraswathi?

Hayagriva, also spelled Hayagreeva is a Horse headed avtar of Lord Vishnu .
Hayagriva is also the name of a horse-headed Danava (descendent of Kashyapa and Danu), who was the first king of danavs.Goddess Sarswati got knowledge from Lord Hayagriva. Recently we celebrated Raakhi Pournami but I did not know that it was the birthday of Lord Hayagriva. A temple totally dedicated to this Lord at Tiruvehendripuram is another amazing fact. Next time when I visit Chenani, I shall try and make it possible to visit Lord Hayagriva too.

Why was Hayagriva avatar taken?

There lived a demon named Hayagriva, the son of Kashyapa and Danu. He did not want the human race to derive the benefits of the four Vedas, therefore, he stole them from Brahma. 

It is said Vishnu was unable to defeat a horse-headed demon because the demon had a boon that he could be killed only by another horse-headed creature. Tired after many battles, Vishnu went to Vaikunth, his abode, and there he rested, wondering of a method of killing the demon, placing his chin on one end of his bow. 
The gods went to him and appealed to him to open his eyes and resume the fight however Vishnu was so deep in meditation that he did not open his eyes. So, Brahma made termites eat the string of the bow. As soon as they severed the taut bowstring, the shaft of the bow snapped straight, and the string swung with such force that it lashed and severed Vishnu’s neck. 
Terrified the gods went to the goddess Laxmi, consort of Vishnu and wondered what they could do. The goddess calmed them down and said nothing happens without a purpose. She advised the gods to behead a horse and place its head on the severed neck of Vishnu. This was done and, Vishnu rose as Hayagriva, the horse headed one. In this form he defeated the horse-headed demon and secured the Vedas. 
This form of Vishnu with a horse’s head is worshipped in many parts of South India. He is depicted holding a conch-shell, a discus, a rosary, and a book. He is associated with knowledge and education and is said to be the form of Vishnu which revealed all knowledge to Saraswathi, goddess of learning. 
Hayagriva is worshipped as a source of wisdom.  The notion of a horse-headed being sharing his wisdom is a recurring theme in Hindu mythology. 

Vishnu Puran Written Update May 22, 2020: Matsya killed Hayagriva and Vishnu took the Vedants. Manu took all 7 Rishis and Shatarupa to form a new life on earth.

Who is the husband of Saraswati?

There are a couple of variations regarding Saraswati’s spouse, but most feature Brahma as her husband. In one tradition, Saraswati married Brahma after she helped him form the universe. Another legend claims that she was first married to Vishnu, the preserver god of the trimurti.