ass.6 - 0 0 0 0 0 0 0 0 The ﬁrst and third columns of A...

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MAT1341 Introduction to Linear Algebra Assignment 6 section 1341A prof. Mike Newman due: Tuesday, November 23 1. Consider the following independent set S of vectors from R 4 : S = 1 0 - 1 1 , 1 1 1 - 1 . a) Is S a spanning set of R 4 ? Justify. [1] b) Extend the set S to a basis of R 4 . [3] 2. Consider the following two matrices A = ± - 1 1 1 0 ² and B = 2 0 5 0 3 0 1 0 3 . a) For each matrix determine its transpose. [2] b) For each matrix determine if it is invertible and if so give its inverse. [4] c) Is B T invertible? If so, then give its inverse. [2] 3. Let A be a 4 × 4 matrix and U a matrix that is row equivalent to A . U = 1 2 0 3 0 0 1 1
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Unformatted text preview: 0 0 0 0 0 0 0 0 . The ﬁrst and third columns of A are respectively a 1 = 2 1-3-1 and a 3 = -1 1-1-1 . a) Give the rank of the matrix A . [1] b) Give the dimension for each of the following subspaces row( A ), col( A ) and ker( A ). [3] c) Give a basis for each of the following subspaces row( A ), col( A ) and ker( A ). [3] 4. Let A and B be two row-equivalent matrices. Are the column spaces of the two matrices necessarily the [2] same? If not, give a counter-example. [ /21]...
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