hi i just wanted a quick explanation of what a symmetric matrix is and what they mean by the quadratic form by the standard basis?

(1)

for example why is this a symmetric matrix

[1 3]

[3 2]

and what is the quadratic form of the matrix by the standard basis?

(2)

also how would i go about figuring out the quadratic form corresponding to the matrix by the standard basis for

[ 0 1 1]

[ 1 3 5]

[ 1 5 0]