Write concise proofs. (a) If dim V > dim W, show that any linear transformation T: V W cannot be one-to-one. (b) Let V be an n-dimensional vector space, and let 0< k < n. If W is an m-dimensional subspace of V and X is an (n-m) -dimensional subspace of V, show that there exists a linear transformation T:V V such that ker T = W and Im T = X.

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Confidence Intervals

When we calculate intervals in probability and statistics, it can be simply termed as finding out a confidence interval. The confidence interval is usually calculated from the data set which is observed. The value of the parameter that needs to be calculated is present here only.