Final answer:
The question involves linear algebra tasks of finding transition matrices between bases and transforming coordinate vectors, but lacks information to provide a specific solution. The answer involves aligning vectors from one basis according to the other basis and using a transition matrix for coordinate transformation.
Step-by-step explanation:
The student's question involves concepts from linear algebra, specifically the transformation of bases and coordinates in the vector space of lower-triangular matrices with zero trace. Without the specific bases provided in the question (since it says 'from to '), it's not possible to give a concrete answer. Nonetheless, the steps to find the transition matrix would generally involve expressing the vectors of one basis in terms of the vectors of the other basis. For part (b), assuming the standard basis and an ordered basis are provided, one would use the transition matrix to convert the coordinate vector of a given vector in the standard basis to the new basis. For part (c), which appears incomplete, no operation can be identified or performed.
To find the transition matrix from one basis to another, generally, you'd align each vector of the new basis as a column in a matrix where each entry is the coefficient of the vectors of the standard basis. If the coordinate vector of a matrix in one basis is known, applying the transition matrix will yield the coordinates in the other basis. However, without the explicit information on what the bases are, or a complete description of part (c), a specific transition matrix or coordinate transformation cannot be provided.
Any correct transition matrix would support operations like determining horizontal and vertical components of vectors and computing scalar products, as mentioned in the other problems you listed.