The advent of high throughput RNA-seq technology allows deep sampling of the transcriptome, making it possible to characterize both the diversity and the abundance of transcript isoforms. Accurate abundance estimation or transcript quantification of isoforms is critical for downstream differential analysis (e.g., healthy vs. diseased cells) but remains a challenging problem for several reasons. First, while various types of algorithms have been developed for abundance estimation, short reads often do not uniquely identify the transcript isoforms from which they were sampled. As a result, the quantification problem may not be identifiable, i.e., lacks a unique transcript solution even if the read maps uniquely to the reference genome. In this article, we develop a general linear model for transcript quantification that leverages reads spanning multiple splice junctions to ameliorate identifiability. Second, RNA-seq reads sampled from the transcriptome exhibit unknown position-specific and sequence-specific biases. We extend our method to simultaneously learn bias parameters during transcript quantification to improve accuracy. Third, transcript quantification is often provided with a candidate set of isoforms, not all of which are likely to be significantly expressed in a given tissue type or condition. By resolving the linear system with LASSO, our approach can infer an accurate set of dominantly expressed transcripts while existing methods tend to assign positive expression to every candidate isoform. Using simulated RNA-seq datasets, our method demonstrated better quantification accuracy and the inference of dominant set of transcripts than existing methods. The application of our method on real data experimentally demonstrated that transcript quantification is effective for differential analysis of transcriptomes.