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# Suppose $u_{1}, u_{2}, \ldots, u_{n}$ belong to a vector space $V$ over a field $K,$ and suppose $P=\left[a_{i j}\right]$ is an $n$ -square matrix over $K .$ For $i... ## Question ###### Suppose$u_{1}, u_{2}, \ldots, u_{n}$belong to a vector space$V$over a field$K,$and suppose$P=\left[a_{i j}\right]$is an$n$-square matrix over$K .$For$i=1,2, \ldots, n,$let$v_{i}=a_{i 1} u_{1}+a_{i 2} u_{2}+\cdots+a_{i n} u_{n}$(a) Suppose$P$is invertible. Show that$\left\{u_{i}\right\}$and$\left\{v_{i}\right\}$span the same subspace of$V$. Hence,$\left\{u_{i}\right\}$is linearly independent if and only if$\left\{v_{i}\right\}$is linearly independent. (b) Suppose$P$is Suppose$u_{1}, u_{2}, \ldots, u_{n}$belong to a vector space$V$over a field$K,$and suppose$P=\left[a_{i j}\right]$is an$n$-square matrix over$K .$For$i=1,2, \ldots, n,$let$v_{i}=a_{i 1} u_{1}+a_{i 2} u_{2}+\cdots+a_{i n} u_{n}$(a) Suppose$P$is invertible. Show that$\left\{u_{i}\right\}$and$\left\{v_{i}\right\}$span the same subspace of$V$. Hence,$\left\{u_{i}\right\}$is linearly independent if and only if$\left\{v_{i}\right\}$is linearly independent. (b) Suppose$P$is singular (not invertible). Show that$\left\{v_{i}\right\}$is linearly dependent. (c) Suppose$\left\{v_{i}\right\}$is linearly independent. Show that$P$is invertible. ## Answers #### Similar Solved Questions 4 answers ##### EXAMPLE 28Not Bad nrouRooklelGoal Apply the kinematlc cquatlons noncer Inltial velocity:freely falllng object20444 Ho AmProblem stone thrown from tne tOp building with an Inltla Velociny stralght upward Initlal helght above the grouna Thc stonc just misses the edge roof on Wdy down snotKn figure. 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