Find the basis of all , and determine its dimension.
The dimension of is which is spanned by role="math" localid="1659419285127" .
Consider the set of all polynomial .
The set role="math" localid="1659419586773" is linear independent set of if there exist constant such that where .
Any polynomial is defined as follows.
Compare the equations bot side as follows.
By the definition of linear independence, the subset is L.I where are linear independent.
Therefore, the set of polynomial is spanned by .
Hence, the dimension of is and spanned by
In Exercises 5 through 40, find the matrix of the given linear transformation with respect to the given basis. If no basis is specified, use standard basis: for ,
for and role="math" localid="1659421462939" for, . For the space of upper triangular matrices, use the basis
Unless another basis is given. In each case, determine whether T is an isomorphism. If T isn’t an isomorphism, find bases of the kernel and image of T and thus determine the rank of T.
12. T from to .
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