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B) Prof: X has two brilliant students Y and Z in his class. He introduces the concept of vector spaces, bases and dimension in his 3rd session_ As an exercise, he gives a vector space V of dimension n and asks Y and Z to find the basis based on the following: Y should start with a set S which has a single non-zero element of V and build the basis by adding vectors to S such that S is linearly independent. The process should be stopped once S has n elements. 2 is asked to start with a set T which
b) Prof: X has two brilliant students Y and Z in his class. He introduces the concept of vector spaces, bases and dimension in his 3rd session_ As an exercise, he gives a vector space V of dimension n and asks Y and Z to find the basis based on the following: Y should start with a set S which has a single non-zero element of V and build the basis by adding vectors to S such that S is linearly independent. The process should be stopped once S has n elements. 2 is asked to start with a set T which has more than elements, say n + k, with k > 0, and remove elements such that T becomes linearly independent. The next day Y and Z bring in their working: Prof: X asks the class if S and T, in the working of Y and Z, the same sets? While student U says yes, student W says need not and Prof: X says that both U and W could be correct Justify the statement of Prof. X with suitable examples of V, S and T. (4
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Let $B$ be a basis for $R^{n}$. Prove that the vectors $v_{1}, v_{2}, \ldots, v_{k}$ form a linearly independent set in $R^{n}$ if and only if the vectors $\left[\mathbf{v}_{1}\right]_{B},\left[\mathbf{v}_{2}\right]_{B}, \ldots,\left[\mathbf{v}_{t}\right]_{B}$ form a linearly independent set in $R^{n}$
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